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In steam, when we say it's 0.9 quality, we mean that 90% is gas and 10% is liquid. But, as for specific volumes, will the specific volume of the gas be the inverse of the mass of the gas over the volume of the whole mixture or of the part of the mixture that the gas occupies? What is that part? In other words, is it correct to say,

$\large{\frac{V_{total}}{m_{(total)}}\ =\ \frac{V_{total}}{m_{(gas)}}+\frac{V_{total}}{m_{(liquid)}}}$

or what? What is the equation?

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No, the specific volume of the steam as for any other property is calculated as follows:

$$ v = v_f + xv_{fg}$$ where $v_f$ is the specific volume of saturated liquid, $v_{fg}$ is the difference between the specific volume of saturated vapor and specific volume of saturated liquid ($v_g - v_f$) (You can obtain them from your steam tables) and $x$ is the dryness factor(steam quality).

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