Question

A \(110-V\) electric hot-water heater warms \(0.1 \mathrm{L} / \mathrm{s}\) of water from 18 to \(30^{\circ} \mathrm{C}\). Calculate the current in amperes that must be supplied to this heater.

Short answer

Answer: The current needed for the heater is approximately 45.65 A.

Step-by-step solution

Determine the temperature difference

As the water is heated from \(18^{\circ}\mathrm{C}\) to \(30^{\circ}\mathrm{C}\), we need to find the temperature difference. To do this, subtract the initial temperature from the final temperature: $$ \Delta T = T_{final} - T_{initial} = 30 - 18 = 12^{\circ}\mathrm{C} $$

Calculate the mass of water per second

We are given the flow rate of water as \(0.1 \mathrm{L/s}\). To find the mass of water per second, we need to convert the volume flow rate to mass flow rate. We can do this knowing the density of water, which is roughly \(1 \mathrm{kg/L}\). Hence, the mass flow rate is: $$ m = 0.1 \: \mathrm{L/s} \times 1 \: \mathrm{kg/L} = 0.1 \: \mathrm{kg/s} $$

Calculate the power required to heat the water

To find the power required, we need to consider the specific heat capacity of water, which is \(c = 4.18 \: \mathrm{J/(g \cdot K)}\). However, we need to convert it to \(\mathrm{J/(kg \cdot K)}\), which is achieved by multiplying by 1000. So, the power needed (P) to heat the water can be calculated using: $$ P = mc\Delta T $$ where \(m\) is the mass flow rate, \(c\) is the specific heat capacity, and \(\Delta T\) is the temperature difference. Plugging in the known values, we get: $$ P = 0.1 \: \mathrm{kg/s} \times 4180 \: \mathrm{J/(kg \cdot K)} \times 12^{\circ}\mathrm{C} = 5021.6 \: \mathrm{J/s} = 5021.6 \: \mathrm{W} $$

Calculate the current needed for the heater

Now, we can use Ohm's law, relating power (P), voltage (V), and current (I): $$ P = IV $$ We can solve for the current I by dividing the power by the voltage: $$ I = \frac{P}{V} $$ Plugging in the values for power (5021.6 W) and voltage (110 V), we obtain: $$ I = \frac{5021.6 \: \mathrm{W}}{110 \: \mathrm{V}} = 45.65 \: \mathrm{A} $$ So, the current that must be supplied to the heater is approximately \(45.65 \: \mathrm{A}\).

Key concepts

Specific Heat Capacity

Specific heat capacity is a measure of the amount of heat energy required to raise the temperature of a unit mass of a substance by one degree Celsius. It is typically denoted by the symbol \( c \) and can have units of energy per mass per degree, such as joules per kilogram per Kelvin (\( J/(kg \cdot K) \)) or calories per gram per degree Celsius (\( cal/(g \cdot ^\circ C) \)).

The specific heat capacity of water is relatively high, which means it takes a lot of energy to change its temperature. This property is crucial in many heating applications, like electric water heaters. In the solved problem, the specific heat capacity of water is used to calculate the energy needed to heat a certain mass flow rate of water from one temperature to another, which is an essential step in determining the power required for the heating process.

The specific heat capacity varies for different materials, which is why knowing the specific heat capacity of the substance you're working with is critical for accurate heat transfer calculations.

Mass Flow Rate

The mass flow rate refers to the amount of mass moving through a cross-section per unit time. It is a vital concept in thermodynamics and fluid dynamics when it comes to heat transfer or other process calculations. For consistency and ease of calculation, it is usually expressed in kilograms per second (\( kg/s \)).

In the context of our problem, the mass flow rate of water is used to ascertain how much water is being heated over time. This figure helps us determine the total amount of heat the heater needs to supply to achieve the desired temperature change. Given the density of water and the volume flow rate, we calculated the mass flow rate as \( 0.1 \text{ L/s} \times 1 \text{ kg/L} = 0.1 \text{ kg/s} \).

Understanding the concept of mass flow rate is essential for engineering tasks related to heating, cooling, and manufacturing processes, as it directly impacts the design and efficiency of the system.

Heat Transfer Equation

The heat transfer equation is a foundational principle in thermodynamics that allows us to calculate the energy required to change the temperature of a substance. The basic formula is \( Q = mc\Delta T \), where \( Q \) is the heat energy transferred, \( m \) represents the mass of the substance, \( c \) is the specific heat capacity, and \( \Delta T \) is the change in temperature.

In the case of the electric water heater problem, this equation was used to determine the power needed to warm the water. By multiplying the mass flow rate of the water by its specific heat capacity and the temperature difference, we obtained the amount of energy per second, or power, required to heat the water. This crucial calculation forms the basis for understanding the energy demands of heating systems and is instrumental in energy management and sustainability practices.

Ohm's Law

Ohm's law is a fundamental principle in electrical engineering that relates the current flowing through a conductor to the voltage across it and the resistance it encounters. In its simplest form, the law states that the current (\( I \)) is equal to the voltage (\( V \)) divided by the resistance (\( R \)). The formula is represented as \( I = V/R \). For power systems, where the resistance is constant, Ohm's law can be used in conjunction with the power formula \( P = IV \), which links power, current, and voltage.

In our textbook solution, Ohm's law is used to find the current that must be supplied to the electric hot-water heater. With the calculated power requirement and the known voltage, we can determine the current using \( I = P/V \). This step is imperative to understanding the electrical demands and ensuring the heater operates safely within its rated capacity. Ohm's law is widely used in all sorts of electrical applications, from simple circuits to complex electrical grids.