I'm afraid I can't give you a precise calculation of the solution: the process is so complicated that, at least in meteorology, we use a semi-empirical formula. I'll describe the process, and give you a reference for the derivation of the semi-empirical formula, which should enable you to solve for your particular case.
The first thing to realize is that, for evaporation into air (or another gas), the vapor pressure will not be the same everywhere. In a micro-layer near the water surface, the vapor pressure will be in equilibrium with the liquid at any instant, with saturation vapor pressure given by the Clausius-Claperyon equation:
e = e0 exp(-.622 L/(R T))where L is the latent heat of vaporization, R is the ideal gas constant for dry air (=287 J/(K kg)) (note that meteorologists define R differently from the R typically used in thermodynamics), and T is the temperature in Kelvin. But in order for more water to be evaporated, the water vapor in this tiny layer must diffuse outward into the surroundings. So the vapor pressure (and thus the rate of evaporation) must be found by solving a diffusion differential equation:
de/dt = K nabla^2 ewhere K is the diffusion constant for water vapor through air. The boundary conditions are e = e(saturation) at the liquid surface, and whatever's appropriate to your situation elsewhere.
But it's even worse than that! Anywhere but in a tightly controlled lab environment, the air will not be still. As it moves, it carries water vapor around. For wind blowing over a land or sea surface, as soon as you get more than a cm or so from the surface, the air becomes a swirling mass of chaotic eddies. This "turbulent boundary layer" extends from tens of meters to a kilometer in altitude, depending on local conditions. We cannot hope to simulate the transport of water vapor through this environment exactly, so we parameterize it, saying that the eddies act to transport water vapor by carrying it from humid regions to dry regions via a diffusive process:
de/dt = Ke nabla^2 ewhere Ke is an eddy diffusivity, which is generally much larger than the molecular diffusivity. Ke increases rapidly as the speed of the mean wind goes up: when the wind blows more rapidly over the ocean, the moisture-carrying eddies are more violent and thus more efficient at carrying vapor away from the sea, thus increasing the evaporation rate.
These ideas lead to a simple, though not entirely precise, formula for the rate of evaporation from a surface, called the "bulk transfer equation for water vapor":
E = -rho Cw |v(z)| (q(z) - q(0))z is the height above the surface at which observations are made (typically 10 m). q is the "specific humidity" of the air, defined as the ratio of water mass to moist air mass for a given volume: q = Mv/M = 0.622 e/p. E is the evaporation rate in kg/(m^2 s), v is mean wind speed, rho is the density of air. q(0) can be assumed to be the saturation specific humidity if we are over a body of water; defining q(0) over land is very tricky. Cw is a dimensionless parameter describing the efficiency of eddies in carrying away moisture. Its value must be determined from observations. It depends to a moderate degree on wind speed and roughness of the surface.
These ideas are presented in "Physics of Climate", by Peixoto and Oort. Section 3.5.1 discusses definitions of atmospheric moisture variables, and section 10.7 discusses evaporation; the derivation of the bulk transfer equation is in 10.7.1. Section 10.7.1 refers to derivations of bulk transfer of heat and momentum, given in sections 10.6 and 10.4.
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