Water is substantially separated from a liquid mixture (e.g., saline water) that includes water using a humidification chamber at a lower pressure and a dehumidification chamber at a higher pressure. A carrier gas is flowed through the humidification chamber; and inside the humidification chamber, the carrier gas directly contacts the liquid mixture to humidify the carrier gas with evaporated water from the liquid mixture to produce a humidified gas flow. The humidified gas flow is then directed through the dehumidification chamber, where water is condensed from the humidified gas flow and collected. The absolute pressure inside the humidification chamber is at least 10% lower than the absolute pressure inside the dehumidification chamber.