Question

In a steam heating system, air is heated by being passed over some tubes through which steam flows steadily. Steam enters the heat exchanger at 30 psia and \(400^{\circ} \mathrm{F}\) at a rate of 15 lbm/min and leaves at 25 psia and \(212^{\circ} \mathrm{F}\). Air enters at 14.7 psia and \(80^{\circ} \mathrm{F}\) and leaves at \(130^{\circ} \mathrm{F}\). Determine the volume flow rate of air at the inlet.

Short answer

Answer: The volume flow rate of the air at the inlet is 0.09607 m³/s.

Step-by-step solution

(Step 1: Write down the ideal gas law.)

(The ideal gas law is \(PV=nRT\), where P is the pressure, V is the volume, n is the number of moles, R is the ideal gas constant, and T is the temperature.)

(Step 2: Find the mass flow rate of the air.)

(We are given the mass flow rate of the steam, but not the air. However, since there is no accumulation or loss of mass in the system, the air and steam must have the same mass flow rate. Therefore, the mass flow rate of the air is also 15 lbm/min.)

(Step 3: Determine the specific volume of the air at the inlet using the ideal gas law.)

(First, let's convert the given temperature and pressure of the air at the inlet into SI units for consistency. \(T_{1} = (80^{\circ} \mathrm{F} - 32) * \frac{5}{9} = 26.67^{\circ} \mathrm{C} = 299.82 K\), and \(P_{1} = 14.7\) psia \(= 101.353 \times 10^{3} \mathrm{Pa}\). The ideal gas constant for air is \(R_{air} = 287 \frac{\mathrm{J}}{\mathrm{kg} \cdot \mathrm{K}}\). Now, we can rearrange the ideal gas law to find the specific volume (\(v_{1}\)) at the inlet: \(v_{1} = \frac{R_{air} \cdot T_{1}}{P_{1}}\). Substituting the values into the equation, we get \(v_{1} = \frac{287 \frac{\mathrm{J}}{\mathrm{kg} \cdot \mathrm{K}} \cdot 299.82 \mathrm{K}}{101.353 \times 10^{3} \mathrm{Pa}} = 0.8475 \frac{\mathrm{m}^3}{\mathrm{kg}}\).)

(Step 4: Calculate the volume flow rate at the inlet.)

(Now that we know the specific volume of the air at the inlet, we can calculate the volume flow rate (\(\dot{V}_{1}\)) by using the mass flow rate of the air (\(\dot{m}_{air}\)): \(\dot{V}_{1} = \dot{m}_{air} \times v_{1}\). First, we need to convert the mass flow rate to the SI unit kg/s: \(\dot{m}_{air} = 15 \frac{\text{lbm}}{\text{min}} \times \frac{0.4536 \text{kg}}{1 \text{lbm}} \times \frac{1 \text{min}}{60 \text{s}} = 0.1134 \frac{\text{kg}}{\text{s}}\). Then, substituting the values into the equation, we get \(\dot{V}_{1} = 0.1134 \frac{\text{kg}}{\text{s}} \times 0.8475 \frac{\mathrm{m}^3}{\mathrm{kg}} = 0.09607 \frac{\mathrm{m}^3}{\mathrm{s}}\).)

(Step 5: Write the final answer.)

(The volume flow rate of the air at the inlet is \(\dot{V}_{1} = 0.09607 \frac{\mathrm{m}^3}{\mathrm{s}}\). This is the final answer.)

Key concepts

Ideal Gas Law

Understanding the ideal gas law is essential when dealing with gases in various thermodynamic processes. It's an equation of state that describes the behavior of ideal gases and is expressed as \(PV = nRT\). In this equation, \(P\) represents pressure, \(V\) is the volume the gas occupies, \(n\) is the number of moles, \(R\) is the universal gas constant, and \(T\) is the absolute temperature (in Kelvins).

This law allows us to predict the behavior of a gas when subjected to changes in temperature, volume, or pressure. For example, in the context of the exercise, we can rearrange the equation to solve for the specific volume \(v\) (volume per unit mass) using the relationship \(v = \frac{V}{m}\). Hence, the ideal gas law is pivotal for determining the specific volume of the air at the inlet, which further allows us to calculate the volume flow rate.

Mass Flow Rate

The mass flow rate is a measure of the amount of mass passing through a given surface per unit time. It's denoted as \(\dot{m}\) and is typically measured in units like \(\frac{kg}{s}\) or \(\frac{lbm}{min}\). For a continuous and steady flow, as seen in the steam heating system in the exercise, the mass flow rate of the air can be assumed to be constant and equal to the mass flow rate of steam, since mass is conserved.

The knowledge of mass flow rate is critical, as it links the amount of substance with its movement through the system. By combining the mass flow rate with specific volume, we can calculate the volume flow rate at the inlet, which is what is required to find the solution for the particular problem posed.

Specific Volume

Specific volume is a thermodynamic property that is the inverse of density. It tells us the volume occupied by a unit mass of a substance, expressed as \(\frac{m^3}{kg}\). In the context of gases, the specific volume plays a crucial role as it is directly related to the gas's temperature and pressure through the ideal gas law.

In our exercise, after converting the air inlet conditions to SI units, we used the ideal gas law to find the specific volume at the inlet. This helped us determine how much space a certain mass of air took up under the given conditions. With this information, calculating the volume flow rate became straightforward—multiplying the specific volume by the mass flow rate gives us the volume flow rate at the inlet.

Thermodynamics

Thermodynamics is the branch of physics dealing with heat and temperature and their relation to energy and work. It involves studying systems and the energy changes that accompany physical and chemical processes. Four fundamental laws govern thermodynamics, which apply regardless of the system's scale or the specific characteristics of the substance.

In the exercise, we apply thermodynamic principles to calculate the volume flow rate of air in a heating system. We relate various properties like temperature, pressure, mass flow rate, and specific volume—all concepts within the purview of thermodynamics. These properties and laws help us describe the behavior of the steam and air in the heating system, allowing us to understand how energy is transferred and transformed within the system.