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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Polytropic process ： ウィキペディア英語版 Polytropic process A polytropic process is a thermodynamic process that obeys the relation: :$p\; v^\; =\; C$ where ''p'' is the pressure, ''v'' is specific volume, ''n'' is the polytropic index (any real number), and ''C'' is a constant. All processes that can be expressed as a pressure and volume product are polytropic processes. Some of those processes (n=0,1,$\backslash gamma,\backslash infty$), are unique. This equation can accurately characterize a very wide range of thermodynamic processes, that range from n=0 to n=$\backslash infty$ which covers, n=0 (isobaric), n=1 (isothermal), n=γ (isentropic), n=$\backslash infty$ (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 polytropic process equation is particularly useful for characterizing expansion and compression processes which include heat transfer. The one restriction is that the process should display a constant energy transfer ratio K during that process: :$K=\backslash delta\; Q/\; \backslash delta\; W=constant$ If it deviates from that restriction it suggests the exponent is not a constant. For a particular exponent, other points along the curve that describes that thermodynamic process can be calculated: :$p\_\}\; =\; p\_v\_^=\; ...\; =\; C$ ==Derivation== The following derivation is taken from Christians.〔Christians, Joseph, "Approach for Teaching Polytropic Processes Based on the Energy Transfer Ratio, ''International Journal of Mechanical Engineering Education'', Volume 40, Number 1 (January 2012), Manchester University Press〕 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 states that the energy added to a system as heat, minus the energy that leaves the system as work, is equal to the change in the internal energy of the system: :$\backslash delta\; q\; -\; \backslash delta\; w\; =\; du$ (Eq. 1) The sign convention is: energy added to the system is counted as positive and energy leaving the system is counted as negative. In the above, $\backslash delta\; q$ is accounted positive as it is the amount of energy entering the system as heat; $\backslash delta\; w$ is accounted negative as it is the amount of energy leaving the system in the form of work done by the system on the environment; while $du$ is the change of the internal energy of the system and will be positive or negative according to the sum of the heat and work terms. Define the energy transfer ratio, :$K\; =\; \backslash frac$ or :$\backslash delta\; q\; =\; K\; \backslash delta\; w$ . For an internally reversible process the only type of work interaction is moving boundary work, given by $\backslash delta\; w\; =\; pdv$. By substituting the above expressions for $\backslash delta\; w$, and $\backslash delta\; q$ into The First Law it can then be written :$(K-1)p\; dv\; =\; c\_\; dT$ (Eq. 1) Consider the Ideal Gas equation of state with the well-known compressibility factor, ''Z'': ''pv = ZRT''. Assume the compressibility factor is constant for the process. Assume the gas constant is also fixed (i.e. no chemical reactions are occurring, hence R is constant). The ''pv = ZRT'' equation of state can be differentiated to give :$p\; dv\; +\; v\; dp\; =\; Z\; R\; dT$ Based on the well-known specific heat relationship arising from the definition of enthalpy, the term ''ZR'' can be replaced by ''cp'' - ''cv''. With these observations the First Law (Eq. 1) becomes :$-\; =\; (1-\; \backslash gamma)K\; +\; \backslash gamma$ where ''γ'' is the ratio of specific heats cp/cv. This equation will be important for understanding the basis of the polytropic process equation. Now consider the polytropic process equation itself: :$p\; v^\; =\; C$ Taking the natural log of both sides (recognizing that the exponent ''n'' is constant for a polytropic process) gives :$\backslash ln\; p\; +\; n\; \backslash ln\; v\; =\; C$ which can be differentiated and re-arranged to give :$n\; =\; -$ By comparing this result to the result obtained from the First Law, it is concluded that when the energy transfer ratio is constant for the process, the polytropic exponent is a constant and therefore the process is polytropic. In fact the polytropic exponent can be expressed in terms of the energy transfer ratio: :$n\; =\; (1-\backslash gamma)K\; +\; \backslash gamma$. where the term $(1-\backslash gamma)$ is negative for an ideal gas. This derivation can be expanded to include polytropic processes in open systems, including instances where the kinetic energy (i.e. Mach Number) is significant. It can also be expanded to include irreversible polytropic processes (see Ref 〔). 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Polytropic process」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.