Polytropic Process

5 Key excerpts on "Polytropic Process"

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  Elements of Energy Conversion

  • Charles R. Russell(Author)
  • 2013(Publication Date)
  • Pergamon

    (Publisher)

  Polytropic Process Any thermodynamic process is termed a Polytropic Process when the pressure-volume relation can be described by pV n = constant Values of n for various conditions are given in Table 2-4. The operation of compressors and turbines where there is some depar-ture from the reversible adiabative process due to friction and heat exchange with the surroundings can be described accurately by the Polytropic Process relations. The value of n must be deter-mined from test data. The work done is W= jpdV = j k -^= -^(ρ,ν,-ρ,ν,) ^-Λ-^Τ,-Τ,) v 1 Vj Also Q the heat added to the system, must equal the sum of the THERMAL PROPERTIES AND RELATIONS 55 work done and the increase in internal energy, or, provided n^l 9 Q = -^liTi-TJ + CvC-TJ = ^ CyiT.-T,) These relations cannot be applied to constant temperature expan-sions where n = l, hence the following relation is derived for a constant temperature process of a perfect gas, where pV = constant = RT W T = j pdV= j ^dV=RTln J* = QT These relations are summarized in Table 2-4. Note that for Polytropic Processes, the constant n may vary from a value greater than y to less than unity. TABLE 2-4 ENERGY RELATIONS FOR PERFECT GASES Process Work done (non-flow) Heat added Constant pressure Constant temperature Constant volume Reversible adiabatic Polytropic y piVz-VJ V 2 NRTn 0 m γ-1 1 n- (PxV 1 -p 2 V 2 ) (PiV 1 -p 2 V 2 ) NCJTt-TJ NC V (T 2 -T Y ) 0 (γ-η) ( W -l ) ( y -l ) (Pi^i-Pi^i) EXAMPLE 2-9. How much work can be done by the expansion of 1 lb of nitrogen at 60 °F from 200 psia to 10 psia in a Polytropic Process where the value of « is 1.2? Solution: Since nitrogen closely approximates a perfect gas under these conditions, the simple gas law can be applied, and

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  Advances in Thermodynamics and Thermal Engineering

  • (Author)
  • 2014(Publication Date)
  • Academic Studio

    (Publisher)

  ____________________ WORLD TECHNOLOGIES ____________________ where p is the pressure, V is volume, n , the polytropic index , is any real number, and C is a constant. This equation can be used to accurately characterize processes of certain systems, notably the compression or expansion of a gas and in some cases liquids and solids. Applicability The equation is a valid characterization of a thermodynamic process assuming that the process is quasistatic and the values of the heat capacities, C p and C V , are almost constant when n is not zero or infinity. (In reality, C p and C V are actually functions of temperature, but are nearly constant within small changes of temperature). Under standard conditions, most gases can be accurately characterized by the ideal gas law. This construct allows for the pressure-volume relationship to be defined for essentially all ideal thermodynamic cycles, such as the well-known Carnot cycle. Note however that there may also be instances where a Polytropic Process occurs in a non-ideal gas. Relationship to ideal processes For certain values of the polytropic index, the process will be synonymous with other common processes. Some examples of the effects of varying index values are given in the table. Variation of polytropic index n Polytropic index Relation Effects n < 0 — An explosion occurs n = 0 pV 0 = p (constant) Equivalent to an isobaric process (constant pressure) n = 1 pV = NkT (constant) Equivalent to an isothermal process (constant temperature) 1 < n < γ — A quasi-adiabatic process such as in an internal combustion engine during expansion, or in vapor compression refrigeration during compression n = γ — γ = is the adiabatic index, yielding an adiabatic process (no heat transferred) — Equivalent to an isochoric process (constant volume) When the index n is between any two of the former values (0, 1, gamma, or infinity), it means that the polytropic curve will bounded by the curves of the two corresponding indices.

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  Flexible Kalina Cycle Systems

  • Tangellapalli Srinivas, N. Shankar Ganesh, R. Shankar(Authors)
  • 2019(Publication Date)
  • Apple Academic Press

    (Publisher)

  The polytropic index plays an important role in this expression as it changes from 0 to ∞, facilitates any thermodynamic process in compression region and expansion region. This section relates the heat and work with polytropic index to understand the energy interactions in the process. We have that the work in Polytropic Process, W 1 − 2 = P 1 V 1 − P 2 V 2 n − 1 = m R (T 1 − T 2) n − 1 From first law of thermodynamics, Q = W + U 2 − U 1 After. simplification, Q = m R (T 1 − T 2) (n − 1) (γ − n γ − 1) (3.37) Q = W (γ − 1 γ − 1) Δ U = Q − W = W (γ − 1 γ − 1) − W = W ((γ − 1 γ − 1) − 1) = W (γ − n − γ + 1 γ − 1[--=PLGO-SEPARATO. R=--]) (3.38) Δ U = W (1 − n γ − 1) (3.39) (i) If n = 0 (constant pressure process), Q = W (γ /(γ − 1)), since γ > γ – 1, Q > W. The heat transfer is more than the work and the increment is equal to the change in internal energy. (ii) If n = 1, Q = W, it is possible with constant internal energy, that is, isothermal process. (iii) If n = γ, Q = 0, that is, it is an adiabatic process. (iv) At n = ∞ (constant volume process), since W = 0 and the second term is equal to infinity and Q ≠ 0. 3.19  Polytropic Process IN OPEN SYSTEM We know that, h = u + P v (3.40) d h = d u + P d v + v d P (3.41) From the first. law, d q = d u + P d v (3.42) d h = d q + v d P (3.43) Since the area under Ts diagram is heat similar to work which is the area under Pv diagram, d h = T d s + v d P (3.44) d q = T d s = d h − ν d P, it is the first law of thermodynamics applied to open system. In place of W, we can find − ν d P. Therefore, in an open system, w = ∫ − v d P (3.45) So the work in polytropic. process, w 1 − 2 = ∫ 1 2 − v d P = − ∫ 1 2 (C P) 1 / n d P ∴ P v n = C = − C 1 / n (P − (1 / n) + 1 − (1 / n) + 1) 1 2 = − C 1 / n (P 2 (n − 1) / n − P 1 (n − 1[--=PL

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  Radial Flow Turbocompressors

  Design, Analysis, and Applications

  • Michael Casey, Chris Robinson(Authors)
  • 2021(Publication Date)
  • Cambridge University Press

    (Publisher)

  In practice, the polytropic efficiency may, like the isentropic efficiency, take different forms depending on the choice of the thermodynamic conditions (total or static conditions) to define the end states of the process. These forms are discussed in the following. Figure 4.7 Isentropic and polytropic efficiency for compressors with γ ¼ 1.4. 121 4.4 Polytropic Efficiency 4.4.4 Static–Static Polytropic Efficiency If it is only the quality of the static enthalpy rise in the machine that is of interest, a so- called static-to-static polytropic efficiency is used. It is defined as the ratio of the polytropic enthalpy rise along a polytropic path between inlet static and outlet static conditions to the actual enthalpy rise along the same path; see Figure 4.9. The differential and integral forms of the static-to-static polytropic efficiency can be written as η ss p ¼ vdp dh (4.31) η ss p ¼ Ð 2 1 vdp Ð 2 1 dh ¼ Ð 2 1 vdp h 2  h 1 ¼ y 12 w s12  ½ c 2 2  c 1 2 ð Þ : (4.32) where ½(c 2 2  c 1 2 ) is the change in kinetic energy from state 1 to state 2. If the change in kinetic energy across the machine is negligible, the static–static polytropic effi- ciency is the true efficiency of the machine. For ideal gases, where the Polytropic Process can be represented by the relation pv n ¼ constant, the aerodynamic work in (4.32) can be derived from integration of vdp, y 12 ¼ ð 2 1 vdp ¼ n n  1 ð Þ p 1 v 1 p 2 p 1   n1 n  1 " # ¼ n n  1 ð Þ RT 1 p 2 p 1   n1 n  1 " # : (4.33) For a perfect gas, the static–static polytropic efficiency can be derived by direct integration of (4.31) as Figure 4.8 The isentropic and polytropic efficiencies in an h-s diagram. 122 Efficiency Definitions for Compressors

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  Thermodynamics and Energy Systems Analysis

  • Lucien Borel, Daniel Favrat, Dinh Lan Nguyen, Magdi Batato(Authors)
  • 2012(Publication Date)
  • EPFL PRESS

    (Publisher)

  Chapter 2 Closed systems and general thermodynamic relations 2.A Polytropic compression Description Consider the system made of a cylinder closed by a piston (Fig. 2.1). The gas contained in the cylinder is compressed from the pressure P 1 to pressure the P 2 . Figure 2.1 This process is represented in Figure 2.2. The gas receives work and passes from the initial state of equilibrium 1 to the final state of equilibrium 2. Hypotheses • The specific volume and the pressure evolves during the process following a polytropic law of type Pv γσ = cst. • The system is thermally insulated. • The change of potential energy is negligible. Data • Initial thermodynamic state of the gas: v 1 = 0.84 m 3 /kg P 1 = 1 bar • Final pressure of the gas: P 2 = 5 bar • Calorific factor relative to the gas: Γ = 0.285 • Polytropic factor relative to the process: γ σ = 1.5 • Mass of the gas: M = 3 kg • Work received by the gas: A + = 450 kJ 22 Polytropic compression Figure 2.2 Questions • Calculate the change of internal energy of the gas. • Calculate the dissipation relative to the process. Solution Change of internal energy of the gas The First Law (1.22) (or the fundamental Equations (2.3)), applied to the process 1-2, give, accounting for the last two hypotheses: ΔU = A + = 450 kJ Dissipation Using the polytropic law indicated in the first hypothesis, we have: v 2 v 1 = P 2 P 1 -1/γσ Note that the above expression, relative to a perfect gas, appears in the Table 8.7 (Vol. I). We deduce the specific volume at the state 2: v 2 = v 1 P 2 P 1 1/γσ = 0.2873 m 3 /kg The initial and final states 1 and 2 being states of equilibrium, the change Δ( ¯ C 2 /2) of specific kinetic energy is null. The first two members of the fundamental Equations (2.3) of a closed system give, according to the previous comment and of the third hypothesis: A - + R = M 2 Z 1 P dv = -M 1 γ σ - 1 (v 2 P 2 - v 1 P 1 )

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