Question

Air is to be heated steadily by an 8 -kW electric resistance heater as it flows through an insulated duct. If the air enters at \(50^{\circ} \mathrm{C}\) at a rate of \(2 \mathrm{kg} / \mathrm{s}\), the exit temperature of air is \((a) 46.0^{\circ} \mathrm{C}\) \((b) 50.0^{\circ} \mathrm{C}\) \((c) 54.0^{\circ} \mathrm{C}\) \((d) 55.4^{\circ} \mathrm{C}\) \((e) 58.0^{\circ} \mathrm{C}\)

Short answer

a) 46.0 \(^{\circ}\)C b) 52.0 \(^{\circ}\)C c) 54.0 \(^{\circ}\)C d) 58.0 \(^{\circ}\)C Given: - Power of the electric heater: 8 kW - Mass flow rate of air: 2 kg/s - Specific heat of air at constant pressure: 1005 J/(kg·K) - Inlet temperature of air: 50 \(^{\circ}\)C Answer: c) 54.0 \(^{\circ}\)C

Step-by-step solution

STEP 1: Identify the given parameters

In this problem, we have: - Power of the electric heater: \(P = 8\ \mathrm{kW} = 8000\ \mathrm{W}\) - Mass flow rate of air: \(\dot{m} = 2\ \mathrm{kg/s}\) - Specific heat of air at constant pressure: \(c_p = 1005\ \mathrm{J/(kg\cdot K)}\) - Inlet temperature of air: \(T_{inlet} = 50\ ^{\circ}\mathrm{C}\)

STEP 2: Calculate the heat transferred per second

Using the given power, we can determine the heat transferred per second: \(Q = P\cdot t = 8000\ \mathrm{W} \cdot t\), Where \(t\) is the time in seconds.

STEP 3: Calculate the temperature increase

The relation between the heat transferred, mass flow rate, specific heat, and temperature increase is given by: \(Q = \dot{m} \cdot c_p \cdot \Delta T\) Now we'll substitute the values for P, m, and cp and solve for the temperature increase, \(\Delta T\): \(8000\ \mathrm{W} \cdot t = 2\ \mathrm{kg/s} \cdot 1005\ \mathrm{J/(kg\cdot K)} \cdot \Delta T\) Divide both sides by 2 kg/s and 1005 J/(kg*K) to isolate \(\Delta T\): \(\Delta T = \frac{8000\ \mathrm{W} \cdot t}{2\ \mathrm{kg/s} \cdot 1005\ \mathrm{J/(kg\cdot K)}}\) \(\Delta T = 4\ \mathrm{K}\)

STEP 4: Calculate the exit temperature of the air

To find the exit temperature, we add the temperature increase to the inlet temperature: \(T_{exit} = T_{inlet} + \Delta T = 50\ ^{\circ}\mathrm{C} + 4\ ^{\circ}\mathrm{K} = 54\ ^{\circ}\mathrm{C}\) Therefore, the correct answer is (c) 54.0 \(^{\circ}\)C.

Key concepts

Energy Transfer

Understanding the concept of energy transfer in thermodynamics is crucial for solving problems involving temperature changes in substances. Energy transfer refers to the process where energy moves from one system or object to another, often resulting in a change of state or temperature.

In our air heating problem, the 8 kW electric resistance heater is an energy source that transfers heat to the air flowing through the duct. The power rating of the heater (8 kW) directly correlates to the rate of energy transfer. It’s important to remember that 1 watt is equivalent to 1 joule per second, which means an 8 kW heater supplies 8000 joules of heat energy per second to the air. This heat energy is what causes the temperature of the air to increase as it absorbs energy.

Specific Heat Capacity

Specific heat capacity is a property that tells us how much heat energy is required to raise the temperature of a unit mass of a substance by one degree Celsius (or one Kelvin). It is usually denoted by the symbol \(c_p\) when pressure is constant, as it most often is in the case of air being heated in a duct.

In our example, the specific heat capacity of air is given as \(1005\ J/(kg\cdot K)\), indicating that every kilogram of air needs 1005 joules of heat to increase its temperature by one Kelvin. Specific heat capacity is vital for calculating the temperature change in materials, as different substances will heat up at different rates even when exposed to the same amount of heat energy.

Mass Flow Rate

The mass flow rate, denoted by \(\dot{m}\), measures the amount of mass passing through a given surface per unit time. In thermodynamics, it’s often expressed in kilograms per second (kg/s) and it’s a key factor when calculating the heat required to change the temperature of flowing substances.

For the air heating scenario, the mass flow rate is \(2\ kg/s\), meaning that each second, 2 kilograms of air passes through the heater. This number is essential for determining how much energy is needed to achieve a desired temperature increase for a continuous flow of air. Knowing the mass flow rate helps us to apply the energy transfer effectively to the specific amount of substance in motion.

Temperature Change Calculation

The temperature change calculation is fundamental for any heating or cooling problem in thermodynamics. It represents the difference between the final and initial temperatures of a system after energy has been transferred to it or removed from it. \

The mathematical relationship for calculating the temperature change (\(\Delta T\)) is established using the formula \(Q = \dot{m} \cdot c_p \cdot \Delta T\), where \(Q\) is the heat added, \(\dot{m}\) is the mass flow rate, and \(c_p\) is the specific heat capacity.

In the exercise we discussed, by rearranging this equation to solve for \(\Delta T\), and knowing the amount of heat energy supplied by the electric heater, we were able to ascertain the temperature increase as \(4\ K\). We then added this increase to the initial temperature of the air to get the exit temperature, confirming that the air's temperature after passing through the heater would be \(54^\circ C\). This step-by-step approach allows students to logically follow the process from energy transfer to the end result, which is the increased air temperature.