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Namely, if I mix a flow of steam that has density $\rho_s$, velocity $v_s$, and pressure $P_s$ with a flow of water that has pressure $P_w$, velocity $v_w$, and density $\rho_w$, how can I find the resulting pressure and density of the mixture?

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In order for two fluid to coexist in anything close to an equilibrium, they must be at the same pressure. After they have been together for a short period of time, they will ultimately come to the same temperature.

If both steam and liquid water exist after equilibrium is achieved, then they must be at what is known as Psat, Tsat. For atmospheric pressure, that temperature would be 100C. The density will be determined by the fraction of the mixture that is steam.

You would need to know the enthalpies of the steam and water going into the mixing as well as the flow rates for each fluid.

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