$($Sandler $4\mathrm{.}31)$ One measure of the thermodynamic efficiency of a complex process $($e$.$g$.,$ an

electrical generation station or an automobile engine$)$ is the ratio of the useful work obtained for

a specified change of state to the maximum useful work obtainable with the ambient temperature

$T_0$ and pressure $P_0.$ Here by useful work, we mean the total work done by the system less the work

done in expansion of the system boundaries against the ambient pressure.

Clearly, to obtain the maximum useful work all pro?cesses should occur reversibly, and all heat

transferred to the surroundings should leave the system at $T_0.$ since heat available at any other

temperature could be used with a Carnot or similar engine to obtain additional work. For a similar

reason, the feed and exit streams in an open process should also be at the ambient conditions.

a$)$ Show that the maximum useful work for a closed system change of state is

$W_{\text{n}\text{e}\text{t}}^{\text{m}\text{a}\text{x}}=A_2-A_1$

Where $A=U+P_{0}V-T_{0}S$ is the closed$-$system availability function.

b$)$ For a steady$-$state or cyclic$-$flow system. show that the maximum useful work is

$W_{\text{n}\text{e}\text{t}}^{\text{m}\text{a}\text{x}}=M(hat(B{)}_1-$widehat$(B{)}_2)$

Where $B=H-T_{0}S$ is the flow availability function and $M$ is the mass that has entered the system

in any convenient time interval in a steady flow process or in one complete cycle in a cyclic

process.

$[$Note that since the inlet and exit temperatures and pressures are equal, hat$(B{)}_1$ will equal hat$(B{)}_2$ and

$W_{\text{n}\text{e}\text{t}}^{\text{m}\text{a}\text{x}}$ will equal zero unless a chemical reaction, $($for example, the burning of coal or gasoline,

a phase change, or some other change in the composition of the inlet and exit streams$)$ occurs.$]$

c$)$ Compute the maximum useful work that can be obtained when $1kg$ of$-$steam undergoes a

closed system change of state from $30$ bar and $600C$ to $5$ bar and $300C$ when the atmospheric

temperature and pressure are as given in the data.

d$)$ A coal$-$fired power generation station generates approximately $2\mathrm{.}2kwh$ of electricity for each

kilogram of coal $($composition given below$)$ burned. Using the availability concept. compute

the thermodynamic efficiency of the station.

Chemical Composition:

$70$ wt$.\%$ Carbon

$15$ wt$.\%$ Water $($liquid$)$

$15$ wt$.\%$ Inorganic matter

Assume that the inorganic matter $($ash$)$ passes through the power station unchanged, that all

the water leaves as vapor in the flue gas, and that all the carbon is converted to carbon dioxide.