Question

Consider a wet concrete patio covered with a thin film of water. At the surface, mass convection of water to air occurs at an average mass transfer coefficient of \(0.03 \mathrm{~m} / \mathrm{s}\). If the air is at $1 \mathrm{~atm}, 15^{\circ} \mathrm{C}$ and 35 percent relative humidity, determine the mass fraction concentration gradient of water at the surface.

Short answer

The mass fraction concentration gradient of water at the surface is approximately 7.5449 * 10^{-7} (mol/m²·s) * V, where V is the volume of the system.

Step-by-step solution

Calculate the saturation pressure

To calculate the saturation pressure of water at \(15^{\circ} \mathrm{C}\), we can use the Antoine equation: \(p_\text{sat} = 10^{A - \frac{B}{T + C}}\), where \(A = 8.07131\), \(B = 1730.63\), and \(C = 233.426\). First, we need to convert the temperature to Kelvin: \(T_K = T + 273.15 = 15 + 273.15 = 288.15 \mathrm{K}\). Then, we can calculate the saturation pressure: \(p_\text{sat} = 10^{8.07131 - \frac{1730.63}{288.15 + 233.426}}\) \(p_\text{sat} = 10^{8.07131 - \frac{1730.63}{521.576}}\) \(p_\text{sat} \approx 1.7000 \mathrm{~kPa}\)

Calculate the partial pressure of water vapor in air

To find the partial pressure of water vapor in the air, we use the given relative humidity (35%) and the saturation pressure: \(p_\text{vapor} = \text{relative humidity} * p_\text{sat} = 0.35 * 1.7000 \mathrm{~kPa} \approx 0.5950 \mathrm{~kPa}\)

Calculate the molar concentration of water vapor

Convert the partial pressure of water vapor (\(p_\text{vapor}\)) to molar concentration using the ideal gas law: \(n_\text{vapor} = \frac{p_\text{vapor} * V}{R * T_K}\), where \(R = 8.314 \mathrm{~J/mol\cdot K}\) is the universal gas constant and \(V\) is the volume. For this exercise, we are not given the volume, but we can still express the molar concentration in terms of volume: \(n_\text{vapor} = \frac{p_\text{vapor}}{R * T_K} * V \approx \frac{0.5950 \mathrm{~kPa} * V}{8.314 \mathrm{~J/mol\cdot K} * 288.15 \mathrm{~K}} \approx 2.5149 * 10^{-5} \frac{\mathrm{mol}}{\mathrm{m^3}} * V\)

Calculate the mass fraction concentration gradient

Finally, we can find the mass fraction concentration gradient (\(\Delta w_\text{vapor}\)) using the mass transfer coefficient (\(k_\text{m} = 0.03 \mathrm{~m/s})\) and the molar concentration of water vapor: \(\Delta w_\text{vapor} = k_\text{m} * n_\text{vapor} \approx 0.03 \mathrm{~m/s} * 2.5149 * 10^{-5} \frac{\mathrm{mol}}{\mathrm{m^3}} * V \approx 7.5449 * 10^{-7} \frac{\mathrm{mol}}{\mathrm{m^2\cdot s}} * V\) Thus, the mass fraction concentration gradient of water at the surface is approximately \(7.5449 * 10^{-7} \frac{\mathrm{mol}}{\mathrm{m^2\cdot s}} * V\).