For psychrometric applications such as those considered in

Chapter 13, Problem 13.112

(choose chapter or problem)

For psychrometric applications such as those considered in Chap. 12, the environment often can be modeled simply as an ideal gas mixture of water vapor and dry air at temperature \(T_{0}\) and pressure \(p_{0}\). The composition of the environment is defined by the dry air and water vapor mole fractions \(y_{\mathrm{a}}^{\mathrm{e}}, y_{\mathrm{v}}^{\mathrm{e}}), respectively.

(a) Show that relative to such an environment the total specific flow exergy of a moist air stream at temperature T and pressure p with dry air and water vapor mole fractions \(y_{a}\) and \(y_{v}\), respectively, can be expressed on a molar basis as

\(\begin{aligned} \overline{\mathrm{e}}_{\mathrm{f}}= & T_{0}\left\{( y _ { \mathrm { a } } \overline { c } _ { p \mathrm { a } } + y _ { \mathrm { v } } \overline { c } _ { p \mathrm { v } } ) \left[\left(\frac{T}{T_{0}}\right)\right.\right. \\ & \left.\left.-1-\ln \left(\frac{T}{T_{0}}\right)\right]+\bar{R} \ln \left(\frac{p}{p_{0}}\right)\right\} \\ & +\bar{R} T_{0}\left[y_{\mathrm{a}} \ln \left(\frac{y_{\mathrm{a}}}{y_{\mathrm{a}}^{\mathrm{e}}}\right)+y_{\mathrm{v}} \ln \left(\frac{y_{\mathrm{v}}}{y_{\mathrm{v}}^{\mathrm{e}}}\right)\right] \end{aligned}\)

where \(\bar{c}_{p \mathrm{a}} \text { and } \bar{c}_{p \mathrm{v}}\) denote the molar specific heats of dry air and water vapor, respectively. Neglect the effects of motion and gravity.

(b) Express the result of part (a) on a per unit mass of dry air basis as

\(\begin{aligned} \overline{\mathrm{e}}_{\mathrm{f}}= & T_{0}\left\{( y _ { \mathrm { a } } \overline { c } _ { p \mathrm { a } } + y _ { \mathrm { v } } \overline { c } _ { p \mathrm { v } } ) \left[\left(\frac{T}{T_{0}}\right)\right.\right. \\ & \left.\left.-1-\ln \left(\frac{T}{T_{0}}\right)\right]+\bar{R} \ln \left(\frac{p}{p_{0}}\right)\right\} \\ & +\bar{R} T_{0}\left[y_{\mathrm{a}} \ln \left(\frac{y_{\mathrm{a}}}{y_{\mathrm{a}}^{\mathrm{e}}}\right)+y_{\mathrm{v}} \ln \left(\frac{y_{\mathrm{v}}}{y_{\mathrm{v}}^{\mathrm{e}}}\right)\right] \end{aligned}\)

where \(R_{\mathrm{a}}=\bar{R} / M_{\mathrm{a}} \text { and } \widetilde{\omega}=\omega M_{\mathrm{a}} / M_{\mathrm{v}}=y_{\mathrm{v}} / y_{\mathrm{a}}\).