Cross-flow membrane module

Liquid-liquid extraction elements are described. A liquid-liquid extraction element includes a first layer pair and a second layer pair disposed adjacent the first layer pair forming a stack of layers. The first layer pair includes a first polymeric microporous membrane, and a first flow channel layer oriented in a first flow direction having a fluid inlet and a fluid outlet disposed on first opposing sides of the extraction element. The second layer pair includes a second polymeric microporous membrane and a second flow channel layer oriented in a second flow direction different than the first flow direction and having a fluid inlet and a fluid outlet disposed on second opposing sides of the extraction element. The first microporous membrane is disposed between the first flow channel and the second flow channel. Methods of extracting a dissolved solute from a first liquid to a second liquid are also described.