THERMODYNAMICS 01. a Certain Quantity of Fluid in a Cylinder Bounded by a Moving Piston Constitutes a A
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Entropy Stable Numerical Approximations for the Isothermal and Polytropic Euler Equations
ENTROPY STABLE NUMERICAL APPROXIMATIONS FOR THE ISOTHERMAL AND POLYTROPIC EULER EQUATIONS ANDREW R. WINTERS1;∗, CHRISTOF CZERNIK, MORITZ B. SCHILY, AND GREGOR J. GASSNER3 Abstract. In this work we analyze the entropic properties of the Euler equations when the system is closed with the assumption of a polytropic gas. In this case, the pressure solely depends upon the density of the fluid and the energy equation is not necessary anymore as the mass conservation and momentum conservation then form a closed system. Further, the total energy acts as a convex mathematical entropy function for the polytropic Euler equations. The polytropic equation of state gives the pressure as a scaled power law of the density in terms of the adiabatic index γ. As such, there are important limiting cases contained within the polytropic model like the isothermal Euler equations (γ = 1) and the shallow water equations (γ = 2). We first mimic the continuous entropy analysis on the discrete level in a finite volume context to get special numerical flux functions. Next, these numerical fluxes are incorporated into a particular discontinuous Galerkin (DG) spectral element framework where derivatives are approximated with summation-by-parts operators. This guarantees a high-order accurate DG numerical approximation to the polytropic Euler equations that is also consistent to its auxiliary total energy behavior. Numerical examples are provided to verify the theoretical derivations, i.e., the entropic properties of the high order DG scheme. Keywords: Isothermal Euler, Polytropic Euler, Entropy stability, Finite volume, Summation- by-parts, Nodal discontinuous Galerkin spectral element method 1. Introduction The compressible Euler equations of gas dynamics ! ! %t + r · (%v ) = 0; ! ! ! ! ! ! (1.1) (%v )t + r · (%v ⊗ v ) + rp = 0; ! ! Et + r · (v [E + p]) = 0; are a system of partial differential equations (PDEs) where the conserved quantities are the mass ! % ! 2 %, the momenta %v, and the total energy E = 2 kvk + %e. -
RECIPROCATING ENGINES Franck Nicolleau
RECIPROCATING ENGINES Franck Nicolleau To cite this version: Franck Nicolleau. RECIPROCATING ENGINES. Master. RECIPROCATING ENGINES, Sheffield, United Kingdom. 2010, pp.189. cel-01548212 HAL Id: cel-01548212 https://hal.archives-ouvertes.fr/cel-01548212 Submitted on 27 Jun 2017 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Distributed under a Creative Commons Attribution - NonCommercial| 4.0 International License Mechanical Engineering - 14 May 2010 -1- UNIVERSITY OF SHEFFIELD Department of Mechanical Engineering Mappin street, Sheffield, S1 3JD, England RECIPROCATING ENGINES Autumn Semester 2010 MEC403 - MEng, semester 7 - MEC6403 - MSc(Res) Dr. F. C. G. A. Nicolleau MD54 Telephone: +44 (0)114 22 27700. Direct Line: +44 (0)114 22 27867 Fax: +44 (0)114 22 27890 email: F.Nicolleau@sheffield.ac.uk http://www.shef.ac.uk/mecheng/mecheng cms/staff/fcgan/ MEng 4th year Course Tutor : Pr N. Qin European and Year Abroad Tutor : C. Pinna MSc(Res) and MPhil Course Director : F. C. G. A. Nicolleau c 2010 F C G A Nicolleau, The University of Sheffield -2- Combustion engines Table of content -3- Table of content Table of content 3 Nomenclature 9 Introduction 13 Acknowledgement 16 I - Introduction and Fundamentals of combustion 17 1 Introduction to combustion engines 19 1.1 Pistonengines.................................. -
High Efficiency and Large-Scale Subsurface Energy Storage with CO2
PROCEEDINGS, 43rd Workshop on Geothermal Reservoir Engineering Stanford University, Stanford, California, February 12-14, 2018 SGP-TR-213 High Efficiency and Large-scale Subsurface Energy Storage with CO2 Mark R. Fleming1, Benjamin M. Adams2, Jimmy B. Randolph1,3, Jonathan D. Ogland-Hand4, Thomas H. Kuehn1, Thomas A. Buscheck3, Jeffrey M. Bielicki4, Martin O. Saar*1,2 1University of Minnesota, Minneapolis, MN, USA; 2ETH-Zurich, Zurich, Switzerland; 3TerraCOH Inc., Minneapolis, MN, USA; 4The Ohio State University, Columbus, OH, USA; 5Lawrence Livermore National Laboratory, Livermore, CA, USA. *Corresponding Author: [email protected] Keywords: geothermal energy, multi-level geothermal systems, sedimentary basin, carbon dioxide, electricity generation, energy storage, power plot, CPG ABSTRACT Storing large amounts of intermittently produced solar or wind power for later, when there is a lack of sunlight or wind, is one of society’s biggest challenges when attempting to decarbonize energy systems. Traditional energy storage technologies tend to suffer from relatively low efficiencies, severe environmental concerns, and limited scale both in capacity and time. Subsurface energy storage can solve the drawbacks of many other energy storage approaches, as it can be large scale in capacity and time, environmentally benign, and highly efficient. When CO2 is used as the (pressure) energy storage medium in reservoirs underneath caprocks at depths of at least ~1 km (to ensure the CO2 is in its supercritical state), the energy generated after the energy storage operation can be greater than the energy stored. This is possible if reservoir temperatures and CO2 storage durations combine to result in more geothermal energy input into the CO2 at depth than what the CO2 pumps at the surface (and other machinery) consume. -
Non-Steady, Isothermal Gas Pipeline Flow
RICE UNIVERSITY NON-STEADY, ISOTHERMAL GAS PIPELINE FLOW Hsiang-Yun Lu A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENT FOR THE DEGREE OF MASTER OF SCIENCE Thesis Director's Signature Houston, Texas May 1963 ABSTRACT What has been done in this thesis is to derive a numerical method to solve approximately the non-steady gas pipeline flow problem by assum¬ ing that the flow is isothermal. Under this assumption, the application of this method is restricted to the following conditions: (1) A variable rate of heat transfer through the pipeline is assumed possible to make the flow isothermal; (2) The pipeline is very long as compared with its diameter, the skin friction exists in the pipe, and the velocity and the change of ve¬ locity aYe small. (3) No shock is permitted. 1 The isothermal assumption takes the advantage that the temperature of the entire flow is constant and therefore the problem is solved with¬ out introducing the energy equation. After a few transformations the governing differential equations are reduced to only two simultaneous ones containing the pressure and mass flow rate as functions of time and distance. The solution is carried out by a step-by-step method. First, the simultaneous equations are transformed into three sets of simultaneous difference equations, and then, solve them by computer (e.g., IBM 1620 ; l Computer which has been used to solve the examples in this thesis). Once the pressure and the mass flow rate of the flow have been found, all the other properties of the flow can be determined. -
Compressible Flow
94 c 2004 Faith A. Morrison, all rights reserved. Compressible Fluids Faith A. Morrison Associate Professor of Chemical Engineering Michigan Technological University November 4, 2004 Chemical engineering is mostly concerned with incompressible flows in pipes, reactors, mixers, and other process equipment. Gases may be modeled as incompressible fluids in both microscopic and macroscopic calculations as long as the pressure changes are less than about 20% of the mean pressure (Geankoplis, Denn). The friction-factor/Reynolds-number correlation for incompressible fluids is found to apply to compressible fluids in this regime of pressure variation (Perry and Chilton, Denn). Compressible flow is important in selected application, however, including high-speed flow of gasses in pipes, through nozzles, in tur- bines, and especially in relief valves. We include here a brief discussion of issues related to the flow of compressible fluids; for more information the reader is encouraged to consult the literature. A compressible fluid is one in which the fluid density changes when it is subjected to high pressure-gradients. For gasses, changes in density are accompanied by changes in temperature, and this complicates considerably the analysis of compressible flow. The key difference between compressible and incompressible flow is the way that forces are transmitted through the fluid. Consider the flow of water in a straw. When a thirsty child applies suction to one end of a straw submerged in water, the water moves - both the water close to her mouth moves and the water at the far end moves towards the lower- pressure area created in the mouth. Likewise, in a long, completely filled piping system, if a pump is turned on at one end, the water will immediately begin to flow out of the other end of the pipe. -
Entropy 2006, 8, 143-168 Entropy ISSN 1099-4300
Entropy 2006, 8, 143-168 entropy ISSN 1099-4300 www.mdpi.org/entropy/ Full paper On the Propagation of Blast Wave in Earth′s Atmosphere: Adiabatic and Isothermal Flow R.P.Yadav1, P.K.Agarwal2 and Atul Sharma3, * 1. Department of Physics, Govt. P.G. College, Bisalpur (Pilibhit), U.P., India Email: [email protected] 2. Department of Physics, D.A.V. College, Muzaffarnagar, U.P., India Email: [email protected] 3. Department of Physics, V.S.P. Govt. College, Kairana (Muzaffarnagar), U.P., India Email:[email protected] Received: 11 April 2006 / Accepted: 21 August 2006 / Published: 21 August 2006 __________________________________________________________________________________ Abstract Adiabatic and isothermal propagations of spherical blast wave produced due to a nuclear explosion have been studied using the Energy hypothesis of Thomas, in the nonuniform atmosphere of the earth. The explosion is considered at different heights. Entropy production is also calculated along with the strength and velocity of the shock. In both the cases; for adiabatic and isothermal flows, it has been found that shock strength and shock velocity are larger at larger heights of explosion, in comparison to smaller heights of explosion. Isothermal propagation leads to a smaller value of shock strength and shock velocity in comparison to the adiabatic propagation. For the adiabatic case, the production of entropy is higher at higher heights of explosion, which goes on decreasing as the shock moves away from the point of explosion. However for the isothermal shock, the calculation of entropy production shows negative values. With negative values for the isothermal case, the production of entropy is smaller at higher heights of explosion, which goes on increasing as the shock moves away from the point of explosion. -
Maximum Thermodynamic Power Coefficient of a Wind Turbine
Received: 17 April 2019 Revised: 22 November 2019 Accepted: 10 December 2019 DOI: 10.1002/we.2474 RESEARCH ARTICLE Maximum thermodynamic power coefficient of a wind turbine J. M. Tavares1,2 P. Patrício1,2 1ISEL—Instituto Superior de Engenharia de Lisboa, Instituto Politécnico de Lisboa, Avenida Abstract Conselheiro Emído Navarro, Lisboa, Portugal According to the centenary Betz-Joukowsky law, the power extracted from a wind turbine in 2Centro de Física Teórica e Computacional, Universidade de Lisboa, Lisboa, Portugal open flow cannot exceed 16/27 of the wind transported kinetic energy rate. This limit is usually interpreted as an absolute theoretical upper bound for the power coefficient of all wind turbines, Correspondence but it was derived in the special case of incompressible fluids. Following the same steps of Betz J. M. Tavares, ISEL—Instituto Superior de Engenharia de Lisboa, Instituto Politécnico de classical derivation, we model the turbine as an actuator disk in a one dimensional fluid flow Lisboa, Avenida Conselheiro Emído Navarro, but consider the general case of a compressible reversible fluid, such as air. In doing so, we are Lisboa, Portugal. obliged to use not only the laws of mechanics but also and explicitly the laws of thermodynamics. Email: [email protected] We show that the power coefficient depends on the inlet wind Mach number M0,andthatits Funding information maximum value exceeds the Betz-Joukowsky limit. We have developed a series expansion for Fundação para a Ciência e a Tecnologia, the maximum power coefficient in powers of the Mach number M0 that unifies all the cases Grant/Award Number: UID/FIS/00618/2019 2 (compressible and incompressible) in the same simple expression: max = 16∕27 + 8∕243M0. -
Thermodynamics for Cryogenics
Thermodynamics for Cryogenics Ram C. Dhuley USPAS – Cryogenic Engineering June 21 – July 2, 2021 Goals of this lecture • Revise important definitions in thermodynamics – system and surrounding, state properties, derived properties, processes • Revise laws of thermodynamics and learn how to apply them to cryogenic systems • Learn about ‘ideal’ thermodynamic processes and their application to cryogenic systems • Prepare background for analyzing real-world helium liquefaction and refrigeration cycles. 2 Ram C. Dhuley | USPAS Cryogenic Engineering (Jun 21 - Jul 2, 2021) Introduction • Thermodynamics deals with the relations between heat and other forms of energy such as mechanical, electrical, or chemical energy. • Thermodynamics has its basis in attempts to understand combustion and steam power but is still “state of the art” in terms of practical engineering issues for cryogenics, especially in process efficiency. James Dewar (invented vacuum flask in 1892) https://physicstoday.scitation.org/doi/10.1063/1.881490 3 Ram C. Dhuley | USPAS Cryogenic Engineering (Jun 21 - Jul 2, 2021) Definitions • Thermodynamic system is the specified region in which heat, work, and mass transfer are being studied. • Surrounding is everything else other than the system. • Thermodynamic boundary is a surface separating the system from the surrounding. Boundary System Surrounding 4 Ram C. Dhuley | USPAS Cryogenic Engineering (Jun 21 - Jul 2, 2021) Definitions (continued) Types of thermodynamic systems • Isolated system has no exchange of mass or energy with its surrounding • Closed system exchanges heat but not mass • Open system exchanges both heat and mass with its surrounding Thermodynamic State • Thermodynamic state is the condition of the system at a given time, defined by ‘state’ properties • More details on next slides • Two state properties define a state of a “homogenous” system • Usually, three state properties are needed to define a state of a non-homogenous system (example: two phase, mixture) 5 Ram C. -
The Effects of Combustion Chamber Design On
THE EFFECTS OF COMBUSTION CHAMBER DESIGN ON TURBULENCE, CYCLIC VARIATION AND PERFORMANCE IN AN SI ENGINE By Esther Claire Tippett B.E.Mech (Hons) University of Canterbury, New Zealand. 1983 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in THE FACULTY OF GRADUATE STUDIES DEPARTMENT OF MECHANICAL ENGINEERING We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA August, 1989 © Esther Claire Tippett, 1989 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of Mechanical Engineering The University of British Columbia Vancouver, Canada Date: ABSTRACT An experimental program of motored and fired tests has been undertaken on a single cylinder spark ignition engine to determine the influence of combustion chamber design on turbulence enhancement in the achievement of fast lean operation. Flow field measurements were taken using hot wire anemometry in the cylinder during motored operation. On line performance tests and in-cylinder pressure data were recorded for the operation of the engine by natural gas at lean and stoichiometric conditions over a range of speed and loads. Squish and squish jet action methods of turbulence enhancement were investigated for six configurations, using a standard bathtub cylinder head and new piston designs incorporating directed jets through a raised wall, a standard bowl-in-piston chamber and an original squish jet design piston. -