Question

Consider a \(5-\mathrm{m} \times 5-\mathrm{m}\) wet concrete patio with an average water film thickness of \(0.3 \mathrm{~mm}\). 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

Answer: No, we cannot determine the drying time for the patio due to the lack of information to calculate the rate of evaporation. Additional data or equations would be necessary to calculate the evaporation rate, taking into account the wind speed, air temperature, and humidity.

Step-by-step solution

Calculate the volume of water on the patio

To find the volume of water on the patio, we can multiply the patio area by the water film thickness. Make sure to convert the thickness from mm to meters. Patio area = \(5\,\text{m} \times 5\,\text{m} = 25\,\text{m}^2\) Water film thickness = \(0.3\,\text{mm} = 0.3 \times 10^{-3}\,\text{m}\) Volume of water = Patio area \(\times\) Water film thickness = \(25\,\text{m}^2 \times 0.3 \times 10^{-3}\,\text{m} = 7.5 \times 10^{-3}\,\text{m}^3\)

Calculate the rate of evaporation

The rate of evaporation depends on several factors, such as the wind speed, air temperature, and humidity. However, this exercise does not provide exact equations to determine the evaporation rate. As a result, we will assume the evaporation rate is directly proportional to the wind speed, air temperature, and relative humidity. To estimate the evaporation rate, we can use the following equation: Evaporation rate (\(\text{m}^3/\text{h}\)) = Constant \(\times\) Wind speed \(\times\) Temperature \(\times\) Humidity Since we don't have a value for the constant, we cannot calculate the evaporation rate. Unfortunately, without the evaporation rate, we cannot determine the drying time.

Conclusion

Due to the lack of information in the provided exercise, the drying time for the patio cannot be determined. To solve this problem, additional data or equations would be necessary to calculate the rate of evaporation, taking into account the wind speed, air temperature, and humidity.

Key concepts

Heat Transfer

Heat transfer plays an essential role in the evaporation process. It involves the movement of thermal energy from one place to another through various means such as conduction, convection, and radiation.

Conduction is the transfer of heat between substances that are in direct contact with each other. Convection, on the other hand, involves the movement of heat by the physical movement of a fluid such as air or water. Finally, radiation is the transfer of heat through electromagnetic waves.

In the context of our problem, when wind blows over a wet concrete patio, it causes convective heat transfer from the air, which provides energy to the water film causing it to evaporate. The higher the temperature of the air and the greater the wind speed, the more energy is available for the water to convert from liquid to vapor, enhancing the evaporation rate.

This convective heat transfer is influenced by the air temperature and velocity; in our exercise, wind is blowing at 50 km/h, which increases the heat transfer to the water film and thus accelerates evaporation.

Mass Transfer

Mass transfer is another key concept integral to evaporation. It refers to the movement of mass from one location, usually within a fluid or between fluids, to another. Evaporation is a mass transfer process in which the molecules of a liquid escape into a gas phase, becoming vapor.

Factors Affecting Mass Transfer in Evaporation

Several factors impact the rate of mass transfer during the evaporation process:

• Concentration Gradient: The difference in vapor concentration between the liquid interface and the surrounding air.
• Wind Velocity: The higher the wind speed, the more quickly vapor is removed from the surface, enhancing the mass transfer.
• Temperature: Higher temperatures typically increase the kinetic energy of molecules, making it easier for them to break into the gas phase.
• Relative Humidity: Low humidity aids evaporation by maintaining a large concentration gradient; the air can accept more water vapor.

In our scenario, wind velocity and relative humidity greatly influence the mass transfer rate of water vapor from the patio surface into the air.

Evaporation Process

The final key concept to understand is the evaporation process itself. Evaporation is the transformation of water from its liquid state into a gas, also known as water vapor. It occurs when molecules at the surface of the liquid have enough kinetic energy to break free from the attraction of other liquid molecules and become a gas.

Evaporation in Open Air

In open-air conditions, like with our concrete patio, the evaporation process can be quite complex. It's not only governed by the temperature and humidity, but also by factors such as:

• Wind speed which aids in dispersing the water vapor away from the surface, allowing more water to evaporate.
• Sunlight, which can provide the heat energy required to increase the kinetic energy of the water molecules.
• The area of the evaporating surface, affecting how much water can evaporate at once.

In the patio drying problem, we assume that heat and mass transfer combine to achieve evaporation. But without specific values or formulas, the rate of evaporation and therefore the time for the patio to dry cannot be accurately determined.

For practical problem solving, real-time measurements or empirical data would help to calculate the constant in the assumed evaporation rate equation, providing students with the ability to predict the drying time under varied conditions.