Understanding Desert Cooler Efficiency and Wet Bulb Temperature

This problem involves understanding the operation of a desert cooler, also known as an evaporative cooler, and its cooling efficiency. Desert coolers cool air by evaporating water, which reduces the air's dry bulb temperature (DBT) and increases its humidity. The theoretical limit of cooling by evaporation is the wet bulb temperature (WBT) of the incoming air.

Key Concepts

• Dry Bulb Temperature (DBT): The temperature of air measured by a standard thermometer. This is the typical temperature reported in weather forecasts.
• Wet Bulb Temperature (WBT): The temperature measured by a thermometer covered in a water-soaked cloth and exposed to moving air. Evaporation of water from the cloth cools the thermometer bulb, and the amount of cooling depends on the air's humidity. Lower humidity leads to more evaporation and a lower WBT.
• Wet Bulb Depression (WBD): The difference between the Dry Bulb Temperature (DBT) and the Wet Bulb Temperature (WBT) ($DBT - WBT$). It indicates the potential for evaporative cooling.
• Cooling Efficiency ($\eta$): For an evaporative cooler, efficiency is defined as the actual temperature drop achieved compared to the maximum possible temperature drop (which is the wet bulb depression of the incoming air).

Calculating Wet Bulb Temperature from Cooler Efficiency

The cooling efficiency ($\eta$) of an evaporative cooler is given by the formula:

$$\eta = \frac{DBT_{in} - DBT_{out}}{DBT_{in} - WBT_{in}}$$

Where:

• $DBT_{in}$ is the initial Dry Bulb Temperature of the air entering the cooler.
• $DBT_{out}$ is the final Dry Bulb Temperature of the air leaving the cooler.
• $WBT_{in}$ is the initial Wet Bulb Temperature of the air entering the cooler (what we need to find).

Applying the Formula to the Problem

We are given the following information:

• Cooling efficiency ($\eta$) = 70% = 0.70
• Initial Dry Bulb Temperature ($DBT_{in}$) = 37°C
• Final Dry Bulb Temperature ($DBT_{out}$) = 30°C

We need to find the initial Wet Bulb Temperature ($WBT_{in}$).

Substitute the given values into the efficiency formula:

$$0.70 = \frac{37°C - 30°C}{37°C - WBT_{in}}$$

Step-by-Step Calculation

1. Calculate the temperature drop in the numerator:

$$37°C - 30°C = 7°C$$

The equation becomes:

$$0.70 = \frac{7°C}{37°C - WBT_{in}}$$

2. Rearrange the equation to solve for the denominator ($37°C - WBT_{in}$):

$$37°C - WBT_{in} = \frac{7°C}{0.70}$$

3. Calculate the value on the right side:

$$\frac{7}{0.70} = 10$$

So, the equation is now:

$$37°C - WBT_{in} = 10°C$$

4. Solve for $WBT_{in}$:

$$WBT_{in} = 37°C - 10°C$$

$$WBT_{in} = 27°C$$

The Wet Bulb Temperature of the air entering the desert cooler is 27°C.