A method of adsorption allows separation of a first fluid component from a fluid mixture comprising at least the first fluid component in an adsorptive separation system having a parallel passage adsorbent contactor with parallel flow passages having cell walls which include an adsorbent material. The method provides for transferring heat from the heat of adsorption in a countercurrent direction along at least a portion of the contactor during adsorption and transferring heat in either axial direction along the contactor to provide at least a portion of the heat of desorption during a desorption step. A carbon dioxide separation process to separate carbon dioxide from flue gas also includes steps transferring heat from adsorption or for desorption along the parallel passage adsorbent contactor.