Thermal Energy Storage Systems and Applications. Ibrahim Dincer

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Figure 1.4 A generalized compressibility chart obtained for 13 fluids (generated through engineering equation solver [EES] software which is used for Z prediction with real gas equations of state). See Ref. [2] for details.

      There are some special cases if either one of P, v, and T is constant. At a fixed temperature, the volume of a given quantity of ideal gas varies inversely with the pressure exerted on it (in some books, this is called Boyle's law), describing compression as

(1.25)

      where the subscripts refer to the initial and final states.

Equation (1.25) is employed by designers in a variety of situations: when selecting an air compressor, for calculating the consumption of compressed air in reciprocating air cylinders, and for determining the length of time required for storing air. Nevertheless, use of Eq. (1.25) may not always be practical due to temperature changes. If temperature increases with compression, the volume of a gas varies directly with its absolute temperature in K as:

(1.26)

      If temperature increases at constant volume, the pressure of a gas varies directly with its absolute temperature in K as:

(1.27)

Equations (1.26) and (1.27) are known as Charles' law. If both temperature and pressure change at the same time, the combined ideal‐gas equation can be written as:

      (1.28)

      For a given mass, the internal energy of an ideal gas can be written as a function of temperature, since c_v0 is constant, as shown below:

(1.29)

      and the specific internal energy becomes

(1.30)

      The enthalpy equation for an ideal gas, based on h = u + Pv, can be written as

      (1.31)

      and the specific enthalpy then becomes

      (1.32)

The entropy change of an ideal gas, based on the general entropy equation in terms of T ds = du + Pdv and T ds = dh − v dP as well as on the ideal‐gas equation Pv = RT, can be obtained in two ways by substituting Eqs. (1.29) and (1.30):

      (1.33)

and

      (1.34)

      For a reversible adiabatic process, the ideal‐gas equation in terms of the initial and final states under Pv^k = constant can be written as:

(1.35)

      where k denotes the adiabatic exponent (the specific heat ratio) as a function of temperature:

      (1.36)

On the basis of Eq. (1.35) and the ideal‐gas equation, the following expression can be obtained:

      (1.37)

      Consider a closed system containing an ideal gas, undergoing an adiabatic reversible process. The gas has constant specific heats. The work can be derived from the first law of thermodynamics (FLT) as follows:

(1.38)

Equation (1.38) can also be derived from the general work relation, W = PdV.

      For a reversible polytropic process, the only difference is the polytropic exponent n which shows the deviation in a log P and log V diagram, leading to the slope. Equations can be rewritten with the polytropic exponent under Pvⁿ = constant as

      (1.39)

      and

      (1.40)

      which results in

      (1.41)

To provide a clear understanding of the polytropic exponent, it is important to show the values of n for four types of polytropic processes for ideal gases (Figure 1.5):