For H2O at a temperature of 300°C and for pressures up to 10,000 kPa (100 bar) calculate values of fi and ϕi from data in the steam tables and plot them vs. P.

Question

For [latex]H_2O[/latex] at a temperature of 300°C and for pressures up to 10,000 kPa (100 bar) calculate values of [latex]f_i and ϕ_i [/latex] from data in the steam tables and plot them vs. P.

Accepted Answer

Equation (10.31) is written twice: first, for a state at pressure P; second, for a low-pressure reference state, denoted by *, both for temperature T:

[latex] G_i \equiv \Gamma_i(T)+R T \ln f_i [/latex] (10.31)

[latex] G_i=\Gamma_i(T)+R T \ln f_i \quad ext { and } \quad G_i^*=\Gamma_i(T)+R T \ln f_i^* [/latex]

Subtraction eliminates [latex]Γ_i(T)[/latex], and yields:

[latex] \ln \frac{f_i}{f_i^*}=\frac{1}{R T}\left(G_i-G_i^* ight) [/latex]

By definition [latex] G_i=H_i-T S_i ext { and } G_i^*=H_i^*-T S_i^* [/latex] ; substitution gives:

[latex] \ln \frac{f_i}{f_i^*}=\frac{1}{R}\left[\frac{H_i-H_i^*}{T}-\left(S_i-S_i^* ight) ight] [/latex] (A)

The lowest pressure for which data at 300°C are given in the steam tables is 1 kPa. Steam at these conditions is for practical purposes in its ideal-gas state, for which [latex] f_i^*=P^*=1 \mathrm{kPa} [/latex]. Data for this state provide the following reference values:

[latex] H_i^*=3076.8 \mathrm{~J} \cdot \mathrm{g}^{-1} \quad S_i^*=10.345 \mathrm{J} \cdot \mathrm{g}^{-1} \cdot \mathrm{K}^{-1} [/latex]

Equation (A) can now be applied to states of superheated steam at 300°C for various values of P from 1 kPa to the saturation pressure of 8592.7 kPa. For example, at P = 4000 kPa and 300°C:

[latex] H_i=2962.0 \mathrm{~J} \cdot \mathrm{g}^{-1} \quad S_i=6.3642 \mathrm{~J} \cdot \mathrm{g}^{-1} \cdot \mathrm{K}^{-1} [/latex]

Values of H and S must be multiplied by the molar mass of water [latex](18.015 g⋅mol^{−1}) [/latex]to put them on a molar basis for substitution into Eq. (A):

[latex] \ln \frac{f_i}{f^*}=\frac{18.015}{8.314}\left[\frac{2962.0-3076.8}{573.15}-(6.3642-10.3450) ight]=8.1917 [/latex]

and

[latex] f_i / f^*=3611.0 [/latex]

[latex] f_i=(3611.0)\left(f^* ight)=(3611.0)(1 \mathrm{kPa})=3611.0 \mathrm{kPa} [/latex]

Thus the fugacity coefficient at 4000 kPa is:

[latex] \phi_i=\frac{f_i}{P}=\frac{3611.0}{4000}=0.9028 [/latex]

Similar calculations at other pressures lead to the values plotted in Fig. 10.2 at pressures up to the saturation pressure P is at [latex] P_i^{ ext {sat }}=8592.7 \mathrm{kPa} [/latex] At this pressure,

[latex] \phi_i^{ ext {sat }}=0.7843 \quad ext { and } \quad f_i^{ ext {sat }}=6738.9 \mathrm{kPa} [/latex]

According to Eqs. (10.39) and (10.41), the saturation values are unchanged by condensation. Although the plots are therefore continuous, they show discontinuities

[latex] f_i^v=f_i^l=f_i^{\mathrm{sat}} [/latex] (10.39)

[latex] \phi_i^v=\phi_i^l=\phi_i^{ ext {sat }} [/latex] (10.41)

in slope. Values of [latex]f_i[/latex] and [latex] ϕ_i[/latex] for liquid water at higher pressures are found using Eq. (10.44), with [latex] V_{i}^{ l}[/latex] equal to the molar volume of saturated liquid water at 300°C:

[latex] f_i=\phi_i^{ ext {sat }} P_i^{ ext {sat }} \exp \frac{V_i^l\left(P-P_i^{ ext {sat }} ight)}{R T} [/latex] (10.44)

[latex] V_i^l=(1.403)(18.015)=25.28 \mathrm{~cm}^3 \cdot \mathrm{mol}^{-1} [/latex]

At 10,000 kPa, for example, Eq. (10.44) becomes:

[latex] f_i=(0.7843)(8592.7) \exp \frac{(25.28)(10,000-8592.7)}{(8314)(573.15)}=6789.8 \mathrm{kPa} [/latex]

The fugacity coefficient of liquid water at these conditions is:

[latex] \phi_i=f_i / P=6789.8 / 10,000=0.6790 [/latex]

Such calculations allow the completion of Fig. 10.2, where the solid lines show how [latex]f_i and ϕ_i[/latex] vary with pressure.

The curve for [latex] f_i [/latex] starts at the origin and deviates increasingly from the dashed line for the ideal-gas state ( [latex] f_{i}^{ ig} = P[/latex] ) as the pressure rises. At [latex] P _{i}^{sat}[/latex] there is a discontinuity in slope, and the curve then rises very slowly with increasing pressure,

indicating that the fugacity of liquid water at 300°C is a weak function of pressure. This behavior is characteristic of a liquid at a temperature well below its critical temperature. The fugacity coefficient [latex]ϕ_i[/latex] decreases steadily from its zero-pressure value of unity as the pressure rises. Its rapid decrease in the liquid region is a consequence of the near-constancy of the fugacity itself.

Question 10.6: For H2O at a temperature of 300°C and for pressures up to 10...

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