Methods and apparatus for distillation using phase change energy

In one embodiment, a method includes moving a first volume of fluid from a region above a heat-transfer element to a region below the heat-transfer element after the first volume of fluid is boiled from a second volume of fluid within the region above the heat-transfer element. The first volume of fluid including an impurity concentration lower than an impurity concentration of the second volume of fluid. The region below the heat-transfer element has a temperature higher than a temperature of the region above the heat-transfer element. The method also includes transferring latent heat from the first volume of fluid to a third volume of fluid on a top surface of the heat transfer element. The latent heat is released when the first volume of fluid condenses.