A numerical model is developed to study the spontaneous liquid flow with an evaporating meniscus in a capillary channel between vertical parallel plates. The time-dependent two-dimensional conservation equations for mass, momentum, and energy are solved using a finite volume scheme. The coupling between the fluid flow, heat transfer, liquid-vapor interface evolution, and recoil pressure induced by liquid evaporation are accounted for. The model predicts the transient evolution of the liquid-vapor interface subject to evaporation, which is established by a temperature difference at the interface between the superheated liquid and the saturated vapor. Prediction and reconstruction of the exact shape of the evaporating meniscus were accomplished by using the arbitrary Lagrangian-Eulerian (ALE) technique with no need for implicit interface tracking schemes. This sharp interface tracking enables direct access to the flow variables and transport fluxes at the meniscus with no need for averaging techniques. The model is used to simulate benzene flow between perfluorocarbon plates with a wall spacing value of 0.7 mm under various liquid superheat conditions. The results indicate that evaporation decreases the equilibrium liquid height compared to the non-evaporating capillary rise. It is also found that the equilibrium capillary height is inversely proportional to the liquid superheat.