Problem 69 - College Physics

Question

\(\bullet\) On-demand water heaters. Conventional hot-water heaters consist of a tank of water maintained at a fixed temperature. The hot water is to be used when needed. The drawback is that energy is wasted because the tank loses heat when it is not in use, and you can run out of hot water if you use too much. Some utility companies are encouraging the use of on- demand water heaters (also known as flash heaters), which consist of heating units to heat the water as you use it. No water tank is involved, so no heat is wasted. A typical household shower flow rate is 2.5 gal \(/ \min (9.46 \mathrm{L} / \mathrm{min}\) ) with the tap water being heated from \(50^{\circ} \mathrm{F}\left(10^{\circ} \mathrm{C}\right)\) to \(120^{\circ} \mathrm{F}\left(49^{\circ} \mathrm{C}\right)\) by the on-demand heater. What rate of heat input (either electrical or from gas) is required to operate such a unit, assuming that all the heat goes into the water?

Step-by-Step Solution

Answer

The required heat input is approximately 25.77 kW.

Step 1: Understand the Problem Statement

We need to calculate the rate of heat input required to heat water from 50°F to 120°F as it flows at 2.5 gallons per minute through an on-demand water heater. First, we need to convert all units to the SI system for consistency.

Step 2: Unit Conversion

Convert the flow rate and temperature change into SI units. The flow rate is 2.5 gal/min, and in SI units this is approximately 9.46 L/min, or 9.46 x 10^-3 m^3/min. The temperature change is from 10°C to 49°C, giving a difference of 39°C.

Step 3: Apply the Heat Equation

The heat input can be calculated using the formula: \[ Q = mc\Delta T \]Where:- \( m \) is the mass flow rate of water in kg/s- \( c \) is the specific heat capacity of water, approximately 4.186 J/g°C- \( \Delta T \) is the temperature change in °C.First, find the mass flow rate: Density of water is approximately 1000 kg/m^3, so the mass flow rate \( m = 9.46 \times 10^{-3} \times 1000 / 60 \approx 0.158 \) kg/s.

Step 4: Calculate the Heat Input

Using the formula:\[ Q = mc\Delta T, \]Substitute in the values we've calculated:\[ Q = 0.158 \times 4.186 \times 39 \approx 25.77 \text{ W} \]Convert joules per minute to watts by dividing by 60:

Step 5: Final Calculation

To get the power in watts: \[ Q = 0.158 \times 4.186 \times 39 \approx 25.77 \text{ kW} \]This is the rate required to heat the water at 2.5 gallons per minute.

Key Concepts

heat input calculation

One of the key tasks is to determine the amount of heat energy required to raise the temperature of water using an on-demand water heater. The heat input is crucial to ensure that water is heated adequately as it flows through the unit. This calculation involves using the formula: \( Q = mc\Delta T \), where:

• \( Q \) is the heat input in joules (J) or watts (W).
• \( m \) is the mass flow rate of the water in kilograms per second (kg/s).
• \( c \) represents the specific heat capacity of water, which is a constant.
• \( \Delta T \) is the change in temperature, measured in degrees Celsius (°C).

This equation helps us understand how much energy is needed to maintain a specific temperature rise in water, which is imperative for efficient heating solutions.

specific heat capacity

Specific heat capacity is a fundamental property of substances, defining how much energy is needed to raise the temperature of a given mass of a substance by one degree Celsius. For water, this value is approximately 4.186 J/g°C. This figure is higher than most other substances, which means water requires more energy to change its temperature.

• Water's high specific heat contributes to its ability to store and transfer large amounts of heat energy without rapid temperature changes.
• This property is important when describing how on-demand water heaters work, as they must efficiently transfer energy to water to heat it instantly as it flows.

Utilizing the specific heat capacity in calculations ensures accurate assessments of the energy needed for temperature adjustments in various applications.

unit conversion

Unit conversion is essential in scientific calculations to ensure consistency and accuracy. The original problem involves converting measurements from the US customary system to the International System of Units (SI). This process involves:

• Converting gallons per minute to cubic meters per second. 2.5 gal/min is approximately 9.46 L/min or 9.46 x 10^-3 m³/min.
• Temperature changes from Fahrenheit to Celsius. For instance, from 50°F to 120°F converts to 10°C to 49°C, giving a temperature change of 39°C.

Unit conversions allow for the application of the heat equation using consistent units, which simplifies calculations and reduces the possibility of errors.

temperature change

To calculate the heat input for on-demand water heaters, understanding temperature change is crucial. A precise value for temperature change helps determine the amount of energy required to raise the water temperature from its initial state to the desired level.

• In this context, the water temperature increases from 10°C to 49°C, making the temperature change 39°C.
• This change is a critical component in the heat equation that helps determine how much energy will be needed.

Calculating accurate temperature changes ensures that water heaters function efficiently, providing the necessary heat output without wastage.

mass flow rate

Mass flow rate measures how much mass of a substance passes through a point per second, which is vital for understanding systems like on-demand water heaters. For water, knowing the mass flow rate allows one to compute how much energy needs to be supplied to heat it as it flows.

• It is calculated by multiplying the volumetric flow rate by the density of water. With the flow rate of approximately 9.46 x 10^-3 m³/min and water's density around 1000 kg/m³, we find a mass flow rate of about 0.158 kg/s.
• This rate helps determine how quickly a water heater must work to supply enough energy to heat the water adequately.

Understanding and calculating the mass flow rate is essential for designing efficient on-demand heating systems and ensuring they meet household water heating demands.