Question

In regions with dry climates, evaporative coolers are used to cool air. A typical electric air conditioner is rated at \(1.00 \times 10^{4} \mathrm{Btu} / \mathrm{h}\) (1 Btu, or British thermal unit \(=\) amount of energy to raise the temperature of \(1 \mathrm{lb}\) water by \(\left.1^{\circ} \mathrm{F}\right) .\) What quantity of water must be evaporated each hour to dissipate as much heat as a typical electric air conditioner?

Short answer

The quantity of water that must be evaporated each hour to dissipate as much heat as a typical electric air conditioner is approximately 10.31 lb/h.

Step-by-step solution

Calculate the rate of energy absorption in Btu/h by the evaporative cooler

To dissipate heat at the same rate as the electric air conditioner, the evaporative cooler must absorb energy at a rate of \(1.00 \times 10^{4}\) Btu/h.

Determine the energy requirement to evaporate 1 lb of water

As given, 1 Btu is the energy needed to increase the temperature of 1 lb of water by \(1^{\circ}\)F. To convert to the energy required to evaporate 1 lb of water, we know that the latent heat of vaporization of water is approximately 970 Btu/lb. Therefore, 970 Btu of energy is needed to evaporate 1 lb of water.

Calculate the quantity of water to be evaporated

Now, let's denote the mass of water to be evaporated per hour as \(m\), measured in pounds (lb). We'll use the relationship between energy, mass, and latent heat of vaporization to find the mass: Energy = mass × latent heat of vaporization \(1.00 \times 10^{4}\) Btu/h = \(m\) × 970 Btu/lb

Solve for the mass of water

To solve for the mass of water, we can simply divide the energy absorbed by the latent heat of vaporization: \(m = \frac{1.00 \times 10^{4}\;\text{Btu/h}}{970\; \text{Btu/lb}}\) \(m \approx 10.31\) lb/h

Final Answer

The quantity of water that must be evaporated each hour to dissipate as much heat as a typical electric air conditioner is approximately 10.31 lb/h.

Key concepts

Latent Heat of Vaporization

The latent heat of vaporization is the energy required to change a substance from a liquid to a gas at its boiling point, without changing its temperature. For water, this is an important concept because it explains how evaporative coolers work. These coolers utilize the principle of latent heat as they evaporate water to cool the surrounding air.
When water evaporates, it absorbs a significant amount of heat energy from the environment. This is because the latent heat of vaporization for water is quite high, approximately 970 British thermal units (Btu) per pound. This means, to turn one pound of water into vapor, you need 970 Btu of energy.
This absorbed energy results in a cooling effect. So, in the context of evaporative cooling, the high latent heat makes water ideal for removing heat from the air and cooling down spaces efficiently. Knowing the latent heat of vaporization is crucial in calculating how much water needs to be evaporated to achieve a certain level of cooling.

British Thermal Unit (Btu)

The British thermal unit (Btu) is an old but still widely used measure of energy, especially in heating and cooling technologies. One Btu represents the amount of heat needed to raise the temperature of one pound of water by one degree Fahrenheit. This makes it a convenient unit for quantifying the amount of energy absorbed or released in heating or cooling processes.
In the context of evaporative coolers, knowing how many Btu are needed helps quantify the energy needed to cool air through water evaporation. For example, an evaporative cooler absorbing heat at a rate of 1.00 x 10,000 Btu/h is equivalent to the operation of a standard air conditioning unit.
Understanding Btus is essential when discussing the efficiency and capacity of cooling systems. It helps compare different types of coolers and select the right one for specific needs, ensuring effective temperature management in varied environments.

Energy Conversion

Energy conversion is the process of transforming one form of energy to another. In evaporative coolers, this involves converting thermal energy absorbed from the environment into latent heat, as water changes from liquid to vapor.
This conversion process plays a key role in determining how much water is necessary for cooling purposes. The energy absorbed in this conversion is the very thing that cools the air. This concept highlights the efficiency of using water vaporization as a passive cooling method.
Calculating energy requirements for the conversion from liquid water to vapor involves understanding both the energy input (initial thermal energy) and the required output (latent heat for vaporization). The formula used is:

• Energy = mass × latent heat of vaporization

This equation is fundamental in determining the cooling capacity of evaporative processes, showcasing how understanding energy conversion is crucial in leveraging natural methods for climate control.