The flow characteristics on a grooved surface depend on the external feeding rate and the overflow from the crest to the valley between the crests. Energetic considerations affecting transition from convex to concave rivulets are presented. The limiting bounds for minimum and maximum flow with full wetting of the groove determine the desired range of practical operation with different groove dimensions. The corresponding average groove-side heat transfer coefficients are presented. Overall heat transfer coefficients for the simultaneous evaporation-condensation process were calculated for various operating conditions and different grooved surfaces. Comparison with experimental data is satisfactory, but nevertheless indicates lack of availability of reliable data for the dynamic solid-liquid contact angle.