Question

Outdoor air \(\left(c_{p}=1.005 \mathrm{kJ} / \mathrm{kg} \cdot^{\circ} \mathrm{C}\right)\) is to be preheated by hot exhaust gases in a cross-flow heat exchanger before it enters the furnace. Air enters the heat exchanger at \(101 \mathrm{kPa}\) and \(30^{\circ} \mathrm{C}\) at a rate of \(0.5 \mathrm{m}^{3} / \mathrm{s}\). The combustion gases \(\left(c_{p}=\right.\) \(\left.1.10 \mathrm{kJ} / \mathrm{kg} \cdot^{\circ} \mathrm{C}\right)\) enter at \(350^{\circ} \mathrm{C}\) at a rate of \(0.85 \mathrm{kg} / \mathrm{s}\) and leave at \(260^{\circ} \mathrm{C}\). Determine the rate of heat transfer to the air and the rate of exergy destruction in the heat exchanger.

Short answer

Answer: To find the rate of heat transfer to the air (\(\dot{Q}\)), we first calculate the mass flow rate of the air using the ideal gas law. Then, we apply the energy balance equation for the heat exchanger, considering the temperature changes and specific heat capacities of both streams. Finally, to calculate the rate of exergy destruction (\(\dot{E}D_\text{hx}\)), we determine the exergy changes for both air and gas streams and apply the exergy balance equation.

Step-by-step solution

Calculate the mass flow rate of air

Use the ideal gas law to find the mass flow rate of air. The ideal gas law is: \(PV = mRT \Rightarrow m = \frac{PV}{RT}\) Given the air enters at \(101 \,\text{kPa}\) and \(30^\circ\mathrm{C}\), and its flow rate is \(0.5 \,\text{m}^3/\text{s}\): \(P = 101\, \mathrm{kPa} = 101\times10^3\, \mathrm{Pa}\) \(V = 0.5 \, \mathrm{m}^3 / \mathrm{s}\) \(T = 30 + 273.15 = 303.15\, \mathrm{K}\) \(R_\text{air} = 0.287 \, \mathrm{kJ} / \mathrm{kg} \cdot \mathrm{K}\) (specific gas constant for air) Now, we can find the mass flow rate of air: \(m_\text{air} = \frac{PV}{RT} = \frac{101\times10^3\mathrm{Pa} \times 0.5\, \mathrm{m}^3/\mathrm{s}}{0.287\, \mathrm{kJ} / \mathrm{kg} \cdot \mathrm{K} \times 303.15\, \mathrm{K}}\)

Calculate the heat transfer rate to the air

Use the energy balance equation for a heat exchanger: \(\dot{Q} = m_\text{air} \cdot c_{p_\text{air}} \cdot \Delta T_\text{air} = m_\text{gas} \cdot c_{p_{\text{gas}}} \cdot \Delta T_\text{gas}\) Solve for the temperature difference of the air: \(\Delta T_\text{air} = \frac{m_\text{gas} \cdot c_{p_{\text{gas}}} \cdot \Delta T_\text{gas}}{m_\text{air} \cdot c_{p_\text{air}}}\) Now, we can find the exit temperature of the air, since we know its inlet temperature is \(30^\circ\mathrm{C}\): \(T_{\text{air},\text{exit}} = T_{\text{air},\text{inlet}} + \Delta T_\text{air}\) Finally, the heat transfer rate to the air is: \(\dot{Q} = m_\text{air} \cdot c_{p_\text{air}} \cdot (T_{\text{air},\text{exit}} - T_{\text{air},\text{inlet}})\)

Calculate the rate of exergy destruction

We first need to find the exergy changes for both air and gas streams: Exergy change for air: \(\Delta E_\text{air} = m_\text{air} \cdot c_{p_\text{air}} \cdot (T_{\text{air},\text{exit}} - T_{\text{air},\text{inlet}}) \cdot \left(1-\frac{T_0}{T_{\text{air},\text{exit}}}\right)\) Exergy change for gas: \(\Delta E_\text{gas} = m_\text{gas} \cdot c_{p_\text{gas}} \cdot (T_{\text{gas},\text{inlet}} - T_{\text{gas},\text{exit}}) \cdot \left(1-\frac{T_0}{T_{\text{gas},\text{inlet}}}\right)\) Now we can write the exergy balance equation for the heat exchanger: \(\dot{E}D_\text{hx} = \Delta E_\text{gas} - \Delta E_\text{air}\) Plug in the calculated expressions for \(\Delta E_\text{air}\) and \(\Delta E_\text{gas}\) to find the rate of exergy destruction in the heat exchanger: \(\dot{E}D_\text{hx} = m_\text{gas} \cdot c_{p_\text{gas}} \cdot (T_{\text{gas},\text{inlet}} - T_{\text{gas},\text{exit}}) \cdot \left(1-\frac{T_0}{T_{\text{gas},\text{inlet}}}\right) - m_\text{air} \cdot c_{p_\text{air}} \cdot (T_{\text{air},\text{exit}} - T_{\text{air},\text{inlet}}) \cdot \left(1-\frac{T_0}{T_{\text{air},\text{exit}}}\right)\)

Key concepts

Cross-Flow Heat Exchanger

A cross-flow heat exchanger is a device where two fluids move perpendicular to each other as they transfer heat. In this type of heat exchanger, one fluid usually moves horizontally and the other vertically, promoting efficient heat transfer due to the crossing flows and increased turbulence. They are extensively used in various applications, including air conditioning systems, power plants, and automotive radiators. Understanding how heat is transferred in a cross-flow heat exchanger requires consideration of the specific heat capacities of the fluids involved, the mass flow rates, and the temperature differences between the inlet and outlet streams.

Mass Flow Rate Calculation

The mass flow rate is a measure of the amount of mass flowing through a given cross-section per unit time and is crucial for the analysis of any heat exchanger. To calculate the mass flow rate of a gas, we often employ the ideal gas law, which relates the volume, pressure, and temperature of an ideal gas with its mass. Given air density can be affected by both temperature and pressure, the ideal gas law allows us to deduce the mass of the air from its flow rate in volume per time, using the specific gas constant for air. This step is fundamental as the mass flow rate directly influences the rate of heat transfer in a heat exchanger.

In educational exercises, these calculations offer students practical understanding of how thermodynamic principles apply to real-world engineering problems.

Ideal Gas Law

The ideal gas law is a well-known equation in thermodynamics, represented by the formula \(PV = nRT\), 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 in Kelvin. For engineering applications, this formula is often rearranged to find the mass \(m\) of the gas: \(PV = mRT\), where \(R\) in this context is the specific gas constant for the gas in question, and \(m\) is the mass of the gas. Constantly we use this equation to connect the physical properties of a gas, enabling us to calculate the mass flow rate, as seen in heat exchanger problems. It's important for students to grasp this concept to solve practical questions regarding gas-related processes.

Energy Balance Equation

The energy balance equation is at the heart of thermodynamics and heat transfer. It states that the rate of heat transfer into a system must equal the rate of heat transfer out of the system, plus any heat generation within the system, minus any heat losses, in steady-state conditions. This principle ensures energy conservation within the system being evaluated.

In the context of a heat exchanger, such as the cross-flow type, the energy balance equation is used to equate the energy entering and leaving with the heat transfer rates of the respective fluids. By evaluating the mass flow rate and specific heat of each fluid alongside their temperature changes, we can determine the total heat transfer rate. This concept is critical for students as it provides the link between theoretical thermodynamics and applied heat exchange processes.

Exergy Destruction Rate

Exergy destruction rate is a measure of irreversibility within a system, a crucial concept in the second law of thermodynamics. It represents the loss of work potential due to energy conversion processes and inherent inefficiencies. In heat exchangers, exergy destruction is caused by the mixing of streams at different temperatures, resulting in entropy generation. Calculating the exergy destruction rate involves determining the changes in exergy of the fluid streams, considering the environmental reference temperature, and utilizing the energy balance for exergy.

This calculation helps identify areas where a heat exchanger's performance can be improved. For students, understanding the exergy destruction rate not only enhances their grasp of energy efficiency and sustainability but also provides valuable insights into the design and operation of thermal systems.