Rate of water evaporation

A mass transfer problem! For water, the mass transfer coefficient is fixed; for a given area, it is related to the difference between the saturated pressure corresponding to the water temperature and the partial pressure of water vapor in the air. The larger it is, the more water vapor evaporates! Put simply, it is related to water temperature and relative humidity in the air. Essence: The difference between the saturated pressure corresponding to the water temperature and the partial pressure of water vapor in the air. It is related to factors such as heat flux density, latent heat of vaporization, and the relative humidity of the evaporation environment. To determine the specific speed, various conditions must be taken into account, as the speed varies under different conditions. The vaporization latent heat can be considered to be related to pressure, and relative humidity can also be understood in terms of the water vapor partial pressure.

The evaporation rate of water is closely related to factors such as the heat transfer rate, the cooling surface area, the evaporation pressure, and the relative humidity of the environment

This post was last edited by sona on 2009-10-14 at 11:57. The factors that affect the rate of water evaporation include area, wind speed, temperature, and relative humidity. The rate of water evaporation is defined as the mass of water that decreases per unit time per unit area. I started by considering the situation where the relative humidity is 100% in order to derive the formula for the evaporation rate. There are several assumptions in this model, namely (1) air molecules are rigid spheres that experience no other forces apart from perfectly elastic collisions with one another. (2) The temperature of the water layer near the liquid surface remains constant throughout, that is, the decrease in temperature of the liquid near the surface caused by evaporation is not taken into account; in other words, the temperature gradient resulting from heat exchange between the interior of the liquid and the surface layer is ignored. (3) When the temperature at the liquid surface remains constant, the rate at which liquid molecules evaporate from the liquid stays fixed, independent of the temperature and relative humidity of the surrounding air. This assumption can be analogized to the escape of photoelectrons, which I will not go into further detail here. We know that when the relative humidity of the air is 100%, the water vapor in the air reaches a saturated state; at this point, the rate of evaporation from the liquid surface is equal to the rate of condensation, and macroscopically, the liquid no longer evaporates. According to the assumptions of the model, the number of water molecules that evaporate from a unit area of the liquid surface per unit time remains constant. Since we do not yet have a model sophisticated enough to accurately describe the state of liquids, it is difficult to determine the evaporation rate of liquids based on their equation of state. From equilibrium physics, we know that at saturation the evaporation rate of water is equal to its condensation rate; therefore, I began by starting from the quantitative relationships in equilibrium to carry out the derivation. The saturated vapor pressure of water at a given temperature and pressure can be measured experimentally. After measuring the saturated vapor pressure of water, the concentration of the saturated vapor of water at that temperature and pressure can be determined using the ideal gas law (referring to the amount of substance of water molecules per unit volume). Assuming that the partial pressure generated by the saturated vapor of water is P0, according to the formula P0V=nRT, the concentration a = n/V = P0/RT. By determining the condensation rate per unit time from the thermostatistical perspective of gases, we can also obtain the evaporation rate of water at that temperature and pressure. To calculate this specific value, we consider air molecules as rigid spheres moving at their respective speeds. Under this assumption, the number of spheres that can collide with the liquid surface per unit time can be regarded as the rate of water condensation, which is equal to the rate of water evaporation. Using Maxwell’s velocity distribution function, we can determine the velocity distribution formula for gas molecules as dN/N = 4π(m/2πkT)^(3/2)exp(-mv^2/2kT)v^2dv. For ease of calculation, the Maxwell speed distribution is rewritten in the form of a velocity distribution formula: dN/N = (m/2πkT)^(3/2)exp(dv dx dy dz). An infinitely long air column with a liquid surface serving as its base area S is taken as the object of study. A coordinate system is established with the direction perpendicular to the liquid surface as the X-axis; within the segment of the column from X to X+ΔX, only those water vapor molecules whose velocity satisfies vx >= X/t can condense into water. By integrating the Maxwell velocity distribution formula over the entire gas column, the amount of water vapor that condenses within time t can be calculated, which corresponds to the amount of liquid that evaporates under those temperature and pressure conditions. Let λ be the evaporation rate of the liquid surface per unit area; then the amount of liquid that evaporates from a surface of area S over a time period t is λSt. The amount of condensation on the liquid surface with area S, as determined using the velocity distribution law, over time t is given by ∫a*S*(m/2πkT)^(3/2)exp(dxdvxdvydvz). The integration limits for this quadruple integral are: x ranges from 0 to infinity, vx ranges from x/t to infinity, vy ranges from 0 to infinity, and vz ranges from 0 to infinity. By first integrating this quadruple integral with respect to vy and vz, it can be transformed into a double integral: ∫a*S*(m/2πkT)^(1/2)expdxdvx. Given that λSt = ∫a*S*(m/2πkT)^(1/2)expdxdvx, it follows that λ = ∫a*(m/2πkT)^(1/2)exp/dxdvx = ∫[∫a*(m/2πkT)^(1/2)expdvx]/t]dx. The time t is arbitrary, and the evaporation rate is independent of time t; therefore, the limit as t approaches 0 can be taken, yielding lim(t→0)[∫a*(m/2πkT)^(1/2)expdvx]/t] = lim(t→0){a*(m/2πkT)^(1/2)exp(x)/t^2}. Substituting this result back into the original integral gives λ = lim(t→0)∫{a*(m/2πkT)^(1/2)exp(x)/t^2}dx = a*(kT/2πm)^(1/2). This formula, λ = a*(kT/2πm)^(1/2), represents the evaporation rate of a liquid. Here, a denotes the molar concentration of water vapor in the air, k is the Boltzmann constant, and m is the mass of the molecules. For ease of use, in the formula λ = a*（kT/2πm）^(1/2), we multiply both the Boltzmann constant k and the mass of the molecule m by the Avogadro constant NA. Using the relationships R=NA*k and M=NA*m, the formula can be simplified to λ = a*（RT/2πM）^(1/2). Here, R represents the ideal gas constant, and M represents the molar mass of the molecule. Once we have the formula for the evaporation capacity of a liquid at a given temperature and pressure, we can discuss the evaporation rate of the liquid under different relative humidities. From a microscopic perspective, as relative humidity decreases, the number of water molecules per unit volume of air decreases; accordingly, the number of molecules that transform from gas to liquid water also decreases. However, the rate at which water evaporates into vapor remains unchanged, so the evaporation rate is greater than the condensation rate. Macroscopically, this results in continuous evaporation of liquid water. If we simply assume that the concentration of water molecules in the air remains constant everywhere, then we can derive a fairly simple formula for evaporation rate in terms of relative humidity. Assuming that we have already determined the evaporation rate of water at that temperature as λ0, the formula for the evaporation rate under different relative humidities is λ = λ0(1-c), where c represents the relative humidity of the air. In reality, in a wind-free environment, the relative humidity at the gas-liquid interface can be considered to be 100%. Then, along the direction of the gas perpendicular to this interface, the relative humidity of the air exhibits a gradient distribution. The presence of this humidity gradient means that the evaporation rate cannot take the simple form of λ = λ0(1-c); therefore, we must take into account the diffusion rate of water molecules. If the concentration gradient distribution of water molecules is known, the diffusion equation can be used to determine the number of gaseous water molecules transported in the direction opposite to the concentration gradient per unit time. The loss of water molecules due to this transport is fully compensated by the evaporation of liquid water; therefore, by determining this transport rate, it is possible to derive the actual evaporation rate equation. The concentration gradient of water vapor molecules is da/dX, and the transport rate of water vapor molecules is -β*da/dX (the negative sign in the formula indicates that the direction of transport is opposite to that of the gradient; β is a constant that represents the diffusion capacity of water vapor). Based on common sense in daily life, we know that clothes dry faster in summer than in winter; in other words, temperature has a significant impact on the rate at which water molecules move. An explanation for this point is obvious from a microscopic perspective: with a constant concentration gradient, an increase in temperature necessarily leads to an increase in the average speed of water molecule movement; as a result, the number of molecules that diffuse across a given interface per unit time also increases. Derive the average velocity of gas molecules in a specific direction in three-dimensional space: Establish a rectangular coordinate system in space, and calculate the average velocity of molecules in the positive x-direction as Vx+ = ∫∫∫ (m/2πkT)^(3/2)exp(vx)dvdxdydz. The limits of integration for this triple integral are vx ranging from 0 to infinity, vy ranging from negative infinity to infinity, and vz also ranging from negative infinity to infinity. The result after integration is Vx+ = (kT/2πm)^(1/2). For ease of calculation, we rewrite this equation as Vx+ = (RT/2πM)^(1/2), where R is the ideal gas constant and M is the molar mass of the molecule. Now, we derive the relationship between the diffusion constant and concentration as well as temperature based on the average velocity of gas molecules in a certain direction (Note: due to the simplicity of the model, the results may be quite inaccurate). Assuming that the molecular concentrations on both sides of a surface with area S are a1 and a2 respectively, then the number of molecules that diffuse across the surface from each side over a time interval t is a1*S*Vx+*t and a2*S*Vx+*t respectively. The net amount of diffusion across the surface is a2*S*Vx+*t – a1*S*Vx+*t = (a2 – a1)*S*Vx+*t. Thus, the amount of diffusion per unit area per unit time is (a2 – a1)*Vx+. Since a2 – a1 = –(da/dx)*dx, when dx approaches 0, the formula (a2 – a1)*Vx+/dx = –(da/dx)*Vx+ becomes the diffusion equation at the interface. By comparing this with the equation –β*da/dX, it can be seen that β = Vx+ = (RT/2πM)^(1/2). If the temperature of water and air is the same, only the concentration gradient at the liquid surface needs to be considered, without having to account for the effect of the temperature gradient on the evaporation rate. In such a situation, with the diffusion coefficient available, the evaporation rate of the liquid can be determined directly once the distribution of the concentration gradient is known; however, the distribution of the concentration gradient is difficult to determine. In fact, we know that beyond a certain distance from the liquid surface, the concentration of water molecules drops to the relative humidity of the entire air; therefore, the concentration gradient of water molecules exists only within distances smaller than this value. Assuming that this value remains constant, the concentration gradient at the same temperature is proportional to a0(1-c), where a0 represents the concentration of saturated water vapor and c represents the relative humidity of the air. Assuming this constant is μ, the concentration gradient can be expressed as μa0(1-c). Thus, using the formula above, the formula for evaporation rate is obtained as λ = μ*a0*(1-c)*√(RT/2πM). Since a0 is also a constant, this formula can be simplified to λ = σ*(1-c)*√(RT/2πM), where σ is a constant. The effect of wind speed on the rate of evaporation arises from the wind at the liquid surface altering the distribution of the humidity gradient in the air surrounding the surface; therefore, by determining the effect of wind speed on the humidity gradient, it is possible to calculate the evaporation rate under that wind speed. No further explanation is needed on this point. I conducted a simple test: at a temperature of 30 degrees Celsius, an air flow speed of 1 m/s, and a relative humidity of 70%, the water evaporation rate was 0.1–0.2 kg/m2·h. This is provided only for reference