Question

A building with an internal volume of \(400 \mathrm{m}^{3}\) is to be heated by a 30 -kW electric resistance heater placed in the duct inside the building. Initially, the air in the building is at \(14^{\circ} \mathrm{C},\) and the local atmospheric pressure is 95 kPa. The building is losing heat to the surroundings at a steady rate of \(450 \mathrm{kJ} / \mathrm{min}\). Air is forced to flow through the duct and the heater steadily by a \(250-\mathrm{W}\) fan, and it experiences a temperature rise of \(5^{\circ} \mathrm{C}\) each time it passes through the duct, which may be assumed to be adiabatic. (a) How long will it take for the air inside the building to reach an average temperature of \(24^{\circ} \mathrm{C} ?\) (b) Determine the average mass flow rate of air through the duct.

Short answer

Based on the given information and calculations, the time needed for the air inside the building to reach an average temperature of 24°C is approximately 3.49 minutes, and the average mass flow rate of air through the duct is 5.97 kg/s.

Step-by-step solution

Calculate the heat loss from the building

Calculate the heat loss from the building in Watts using the given rate of 450 kJ/min. Heat loss rate, \(Q_{loss} = 450,000 \frac{\text{J}}{\text{min}} \times \frac{1 \text{min}}{60 \text{s}} = 7500\, \text{W}\)

Calculate the heat added by the electric resistor to the air

The electric resistance heater has a power of 30 kW. We also need to account for the heat added by the 250-W fan. The heat added by the resistance heater, \(Q_{heater} = 30,000\, \text{W}\) The heat added by the fan, \(Q_{fan} = 250\, \text{W}\) The total heat added to the air, \(Q_{added} = Q_{heater} + Q_{fan} = 30,250\, \text{W}\)

Calculate the net heat transfer rate

Subtract the heat loss rate from the total heat added to the air. Net heat transfer rate, \(Q_{net} = Q_{added} - Q_{loss} = 30,250\, \text{W} - 7,500\, \text{W} = 22,750\, \mathrm{W}\)

Calculate the energy required to increase the air temperature from 14°C to 24°C

Using the specific heat for air, \(c_p = 1005 \frac{\text{J}}{\text{kg} · \text{K}}\), and assuming constant pressure, the energy required can be calculated as follows: \(Q = mc_p(T_f - T_i)\), where \(Q\) is the amount of energy, \(m\) is the total mass of the air in the building, \(T_f = 24^{\circ}\mathrm{C}\), and \(T_i = 14^{\circ}\mathrm{C}\). Since the air density is \(\rho_a = 1.184 \frac{\text{kg}}{\text{m}^{3}}\), we can approximate the mass of air in the building as: \(m = \rho_a \times V = 1.184 \frac{\text{kg}}{\text{m}^{3}} \times 400 \text{m}^{3} = 474 \mathrm{kg}\) Now, calculate the energy required, \(Q\): \(Q = (474\, \text{kg})(1005\, \frac{\text{J}}{\text{kg}\cdot \text{K}})(10\, \text{K}) = 4,763,700\, \text{J}\)

Estimate the time taken to reach the desired temperature

Divide the total energy required by the net heat transfer rate to obtain the time: \(t = \frac{Q}{Q_{net}} = \frac{4,763,700\, \text{J}}{22,750\, \mathrm{W}} = 209.34\, \text{s}\) Converting to minutes, we have \(t = 3.49\, \text{min}\)

Determine the mass flow rate of air through the duct

Using the temperature rise per pass through the duct, \(\Delta T = 5^{\circ} \mathrm{C}\), the power of the heater, and the specific heat capacity of air, we can find the mass flow rate of air through the duct. \(Q_{heater} = \dot{m}c_{p}\Delta T \Rightarrow \dot{m} = \frac{Q_{heater}}{c_p \Delta T} = \frac{30,000\, \text{W}}{1005\, \frac{\text{J}}{\text{kg}\cdot \text{K}} \times 5\, \text{K}} = 5.97\, \frac{\text{kg}}{\text{s}}\) The time taken for the air inside the building to reach an average temperature of 24°C is approximately 3.49 minutes. The average mass flow rate of air through the duct is 5.97 kg/s.

Key concepts

Heat Transfer Calculations

Understanding heat transfer calculations is essential for solving a wide range of thermodynamics problems, including the heating of buildings. Heat transfer is a phenomenon involving the movement of thermal energy from one object or material to another, driven by a temperature difference between them. It can occur through various mechanisms such as conduction, convection, and radiation.

Application in the Building Heating Problem

For our particular exercise, the heat transfer we're concerned with is the amount of thermal energy added by a heater and fan to the air inside a building, as well as the rate at which the building loses heat to the surrounding environment. Heat transfer calculations help us estimate the time required for the air temperature within the building to reach a certain level by using an energy balance approach.

Specific Heat Capacity

Specific heat capacity, denoted as 'cp', is a property that defines how much heat energy is required to raise the temperature of a unit mass of a substance by one degree of temperature. It is usually expressed in units of joules per kilogram per Kelvin (J/kg·K). The specific heat capacity is crucial when we want to determine the amount of energy required for temperature change in a material.

Role in Our Heating Problem

In the exercise given, specific heat capacity allows us to calculate the energy needed to increase the air temperature inside the building. Knowing the specific heat capacity of air, we can determine the total heat energy required to raise the temperature of the entire mass of air by 10°C from 14°C to 24°C.

Mass Flow Rate

The concept of mass flow rate is the measure of the amount of mass passing through a given surface per unit time. It is a critical parameter in fluid mechanics and thermodynamics, measured in kilograms per second (kg/s) in the SI system. Understanding the mass flow rate is essential when analyzing systems where fluid, such as air or water, is being moved or circulated.

Determining Mass Flow Rate in Ducts

For the problem at hand, determining the mass flow rate of air through the duct is significant because it relates to the effectiveness of the heating system. The mass flow rate is derived using the heat provided by the heater, the specific heat capacity of air, and the temperature rise of the air after passing through the duct. This value ultimately helps us understand how efficiently the heater can maintain the desired indoor temperature amidst continuous heat loss.

Energy Balance

Energy balance is a fundamental concept in thermodynamics stating that energy cannot be created or destroyed, only transformed from one form to another. In a steady-state system, the energy balance involves equating the energy entering the system to the energy leaving it plus any change in the energy stored in the system.

Calculating Energy Balance for the Heating System

For the issue presented in the exercise, we establish an energy balance by equating the total heat energy added to the air by the heater and fan to the sum of the heat lost to the surroundings and the energy needed to raise the air temperature within the building. This energy balance makes it possible to calculate both the time it takes for the building to reach a designated temperature and the average mass flow rate, which indicates the performance of the heating system in real-time conditions.