Problem 49 - Physics

Question

An electric hot water heater takes in cold water at \(13.0^{\circ} \mathrm{C}\) and delivers hot water. The hot water has a constant temperature of \(45.0^{\circ} \mathrm{C}\) when the "hot" faucet is left open all the time, and the volume flow rate is \(5.0 \times 10^{-6} \mathrm{~m}^{3 / \mathrm{s}}\). What is the minimum power rating of the hot water heater?

Step-by-Step Solution

Answer

The minimum power rating of the hot water heater is 669.76 W.

Step 1: Understanding the Problem

We need to find the minimum power required to heat water from \(13.0^{\circ} \mathrm{C}\) to \(45.0^{\circ} \mathrm{C}\) at a flow rate of \(5.0 \times 10^{-6} \mathrm{~m}^{3} / \mathrm{s}\). This involves calculating the energy needed to heat the water.

Step 2: Calculate Mass Flow Rate

To find the mass flow rate, we use the density of water, which is approximately \(1000 \mathrm{~kg} / \mathrm{m}^{3}\). The mass flow rate \( \dot{m} \) is given by: \[ \dot{m} = \text{density} \times \text{volume flow rate} = 1000 \times 5.0 \times 10^{-6} \mathrm{~kg/s} = 0.005 \mathrm{~kg/s} \]

Step 3: Calculate Energy Required per Second

We need to calculate the amount of energy required to heat 0.005 kg of water by \(32^{\circ} \mathrm{C}\) (from \(13^{\circ} \mathrm{C}\) to \(45^{\circ} \mathrm{C}\)). The specific heat capacity \(c\) of water is \(4,186 \mathrm{~J/(kg \cdot °C)}\). The energy required is: \[ Q = \dot{m} \times c \times \Delta T = 0.005 \times 4186 \times 32 \]

Step 4: Calculate Power

Power \(P\) is the energy per unit time (in this case, per second). Using the energy calculated: \[ P = Q = 0.005 \times 4186 \times 32 = 669.76 \mathrm{~W} \] The minimum power rating of the water heater is therefore \(669.76 \mathrm{~W}\).

Key Concepts

Understanding the Specific Heat Capacity of Water

Water is an excellent medium for thermal energy due to its high specific heat capacity. This term refers to the amount of energy in joules needed to raise the temperature of one kilogram of a substance by one degree Celsius. For water, this value is notably high at 4,186 J/(kg°C).
This means water can store a lot of heat energy without its temperature rising too quickly, which is crucial for applications like heating systems. When heating water from a lower to a higher temperature, it's essential to consider this property to calculate the energy requirement effectively.
The formula to find the energy required to heat a water mass is: \[ Q = m \times c \times \Delta T \] where:

• \(Q\) is the heat energy in joules,
• \(m\) is the mass of the water,
• \(c\) is the specific heat capacity,
• \(\Delta T\) is the temperature change.

Understanding this concept is key to solving problems involving heating water, such as determining the power necessary for a water heater.

Exploring the Mass Flow Rate of Water

The mass flow rate indicates the amount of mass moving through a system per unit of time, typically expressed in kilograms per second (kg/s). For liquids like water, the mass flow rate can be calculated using the formula: \[ \dot{m} = \rho \times Q \] where:

• \(\dot{m}\) is the mass flow rate,
• \(\rho\) represents the density of the liquid (for water, it's about 1,000 kg/m³),
• \(Q\) is the volumetric flow rate.

In scenarios where water is constantly being heated and used, like through a tap or water heater, understanding the mass flow rate helps calculate the energy required and thus the power needed.
For example, if the volumetric flow rate is given as \(5.0 \times 10^{-6} \mathrm{~m}^{3}/\mathrm{s}\), multiplying by the density of water gives a mass flow rate of 0.005 kg/s. This is crucial because the energy required to heat water is directly proportional to the amount of water flowing per second.

Temperature Change in Water

Temperature change, denoted as \(\Delta T\), is the difference between the final and initial temperatures in a thermal process. In practical applications, calculating \(\Delta T\) helps ascertain how much heating or cooling is necessary.
For the problem of heating water in a hot water heater, the initial temperature is \(13.0^{\circ}\mathrm{C}\) and the final temperature is \(45.0^{\circ}\mathrm{C}\). Thus, the temperature change is calculated as: \[ \Delta T = T_{\text{final}} - T_{\text{initial}} = 45.0^{\circ}\mathrm{C} - 13.0^{\circ}\mathrm{C} = 32^{\circ}\mathrm{C} \]
Recognizing the importance of this change in temperature is crucial because it directly influences the amount of energy needed for heating.
This information becomes particularly important when combined with the specific heat capacity of water and the mass flow rate, as it allows accurate calculation of power requirements for devices like water heaters.