Fundamentals of Heat Engines. Jamil Ghojel

equation shows that T_L can never be zero or less than zero, for if T_L were equal to zero, the thermal efficiency of the heat engine would be 1, or 100%, which would be a violation of the second law. Also, if T_L were less than zero, the thermal efficiency would be greater than 100%, which would mean a reversal of the direction of Q_L, and heat would be drawn from the low‐temperature source. This also would be a violation of the second law. So, theoretically, the absolute zero of temperature could never be reached.

If a Carnot engine is operated between a high‐temperature source at the temperature of boiling water T_H and a low‐temperature sink at the temperature of melting ice T_L, and the amount of heat received Q_steam and rejected Q_ice were measurable accurately, the ratio Q_steam/Q_ice would be equal to 1.3662. Since Q_steam/Q_ice = T_steam/T_ice,

(1.90)

      If the temperature range T_steam − T_ice is arbitrarily divided into 100 equal divisions (call it 100 °C), a second equation can be obtained

(1.91)

      The simultaneous solution of Eqs. (1.90) and (1.91) yields the temperature of boiling water and melting ice on a thermodynamic scale:

      If the temperature range T_steam − T_ice is divided into 180 equal divisions (call it degrees Fahrenheit, F), we get

      (1.92)

      The determination of a thermodynamic scale using reversible Carnot engine as described here is practically impossible. To determine where the number 1.3662 comes from, we start with the equation of state for a perfect gas at constant volume and constant pressure, which yields

      (1.93)

      Next, constant‐volume and constant‐pressure thermometers are placed alternately in boiling water and melting ice, and the volume and pressure ratios of the gases in the devices (carbon dioxide CO₂, hydrogen H₂, and helium He) are measured at an initial pressure. If the procedure is repeated for different initial pressures and the results plotted versus pressure, straight lines are obtained, which converge at zero absolute pressure to a value of 1.3662, which must be the ratio of absolute temperature of boiling water and melting ice.

      Currently, the international temperature scale (ITS‐90) is the standard used to represent the thermodynamic (absolute) temperature scale as closely as possible with improved accuracy over gas thermometry.

      1.3.9 Third Law of Thermodynamics

      This law, based on empirical evidence, postulates that absolute entropy of a pure crystalline substance in complete internal equilibrium is zero at temperature zero degree absolute. The third law allows the determination of absolute entropies from thermal data.

      The entropy change of a gas on molar basis is

The entropy change for process 1–2 is then

(1.94)

      If the specific heat C_p is assumed constant, Eq. (1.94) becomes

      (1.95)

      More accurate results can be obtained if the variability of specific heat with temperature is accounted for. Taking the absolute zero as the reference temperature, can be defined as

(1.96)

      The values of are usually calculated for different temperatures and can be found in tabular form as s⁰ = f(T) in most thermodynamic reference books. Using Eq. (1.96), we can write

(1.97)

      Substituting Eq. (1.97) in Eq. (1.94), we obtain

(1.98)

      As Eq. (1.98) shows, entropy changes with both temperature and pressure. When using Eq. (1.98) in chemical reactions, the pressure ratio in the last term is replaced by the mole concentration of each substance.

      The data for some commonly used gases in heat engine practice such as CO₂, CO, H₂O, H₂, O₂, and N₂ are tabulated in Appendix A. They can be calculated to a very high degree of accuracy for the temperature range 100–6000 K by third‐order logarithm functions of the form

(1.99)

The coefficients of correlation Eq. (1.99) for six gases are given in Table 1.3.

Table 1.3 Coefficients of Eq. (1.99) for the calculation of

of combustion products, p_ref = 0.1 MPa T_ref = 0 K.

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	, kJ/kmole.K
Coefficient	CO2	CO	H2O	H2	O2	N2