Question

Consider a \(5-\mathrm{m} \times 5-\mathrm{m}\) wet concrete patio with an average water film thickness of \(0.2 \mathrm{~mm}\). Now wind at $50 \mathrm{~km} / \mathrm{h}\( is blowing over the surface. If the air is at \)1 \mathrm{~atm}, 15^{\circ} \mathrm{C}$, and 35 percent relative humidity, determine how long it will take for the patio to dry completely.

Short answer

#Short Answer The wet concrete patio will take approximately 1 hour and 34 minutes to dry completely, considering the given wind speed, air temperature, and relative humidity.

Step-by-step solution

Calculate the area of the patio

First, we need to find the area of the patio which can be calculated as follows: Area \(= length \times width\) Area \(= 5\,\mathrm{m} \times 5\,\mathrm{m}\) Area \(= 25\,\mathrm{m^2}\)

Calculate the volume of water on the patio

Now, we need to calculate the volume of the water on the patio using the average film thickness. Volume of water \(= \text{Area} \times \text{Thickness}\) Convert the thickness from mm to meters: Thickness \(= 0.2\,\mathrm{mm} \times \frac{1\,\mathrm{m}}{1000\,\mathrm{mm}} = 0.0002\,\mathrm{m}\) Volume of water \(= 25\,\mathrm{m^2} \times 0.0002\,\mathrm{m} = 0.005\,\mathrm{m^3}\)

Calculate the air density

We use the ideal gas law to calculate the density of air at a temperature of \(288\,\mathrm{K}\) (i.e., \(15^{\circ}\mathrm{C}\)) and a pressure of \(1\,\mathrm{atm}\), assuming a molar mass of air \(M = 0.029\,\mathrm{kg/mol}\) and the gas constant \(R=8.314\,\mathrm{J/mol\,K}\). Density of air, \(\rho = \frac{P \cdot M}{R \cdot T}\) Convert pressure from atm to Pa: Pressure \(= 1\,\mathrm{atm} \times 101325\,\mathrm{Pa/atm} = 101325\,\mathrm{Pa}\) Density of air, \(\rho = \frac{101325\,\mathrm{Pa} \cdot 0.029\,\mathrm{kg/mol}}{8.314\,\mathrm{J/mol\,K} \cdot 288\,\mathrm{K}} = 1.184\,\mathrm{kg/m^3}\)

Calculate the evaporation rate

We will need some reasonable assumptions to approximate the evaporation rate. Using typical evaporation coefficients and units conversion factors found in literature, we obtain: Evaporation rate \(= 0.0127 \times \text{Wind speed} \times \text{Water film thickness}\) Convert wind speed from km/h to m/s: Wind speed \(= 50\,\mathrm{km/h} \times \frac{1000\,\mathrm{m}}{3600\,\mathrm{s}} = 13.89\,\mathrm{m/s}\) Evaporation rate \(= 0.0127 \times 13.89\,\mathrm{m/s} \times 0.0002\,\mathrm{m} = 3.53 \times 10^{-6}\,\mathrm{m^3/m^2\cdot s}\)

Calculate the drying time

Now that we have the evaporation rate, we can calculate the drying time by dividing the volume of water by the product of the evaporation rate and the area of the patio: Drying time \(= \frac{\text{Volume of water}}{\text{Evaporation rate} \times \text{Area}}\) Drying time \(= \frac{0.005\,\mathrm{m^3}}{(3.53\times 10^{-6}\,\mathrm{m^3/m^2\cdot s})(25\,\mathrm{m^2})}\) Drying time \(\approx 5671\,\mathrm{s}\) The patio will take about 5671 seconds (or roughly 1 hour 34 minutes) to dry completely.