Question

In large gas-turbine power plants, air is preheated by the exhaust gases in a heat exchanger called the regenerator before it enters the combustion chamber. Air enters the regenerator at \(1 \mathrm{MPa}\) and \(550 \mathrm{K}\) at a mass flow rate of \(800 \mathrm{kg} / \mathrm{min}\). Heat is transferred to the air at a rate of \(3200 \mathrm{kJ} / \mathrm{s}\). Exhaust gases enter the regenerator at \(140 \mathrm{kPa}\) and \(800 \mathrm{K}\) and leave at \(130 \mathrm{kPa}\) and \(600 \mathrm{K}\). Treating the exhaust gases as air, determine ( \(a\) ) the exit temperature of the air and \((b)\) the mass flow rate of exhaust gases.

Short answer

Based on the given information and calculations, we can determine the following: a) The exit temperature of the air after leaving the regenerator is 789.05 K. b) The mass flow rate of the exhaust gases is 954.23 kg/min.

Step-by-step solution

Calculate the specific enthalpy of the air entering the regenerator

We are given the initial pressure and temperature of the air (1 MPa and 550 K), and we can use the ideal gas law (\(pv = RT\)) to find the specific volume of the air. Using the ideal gas constant for air (\(R = 0.287 \mathrm{kJ/kgK}\)), we can get: \(v_1 = \frac{RT_1}{P_1} = \frac{0.287 \times 550}{1000} = 0.15785 \mathrm{m^3/kg}\) Now, we can compute the enthalpy of air using the specific heat of air at constant pressure (\(c_p = 1.005 \mathrm{kJ/kgK}\)): \(h_1 = c_p (\underline{T_1}) = 1.005 \times 550 = 552.75 \mathrm{kJ/kg}\)

Calculate the air enthalpy after it has been heated

Since we are given the heat transfer rate (3200 kJ/s) and the mass flow rate of the air (800 kg/min), we can calculate the heat absorbed per unit mass: \(q = \frac{3200}{(800/60)} = 240 \mathrm{kJ/kg}\) Applying the energy balance equation on a per mass basis, and considering only heat input and output, gives us: \(h_2 = h_1 + q = 552.75 + 240 = 792.75 \mathrm{kJ/kg}\)

Calculate the exit temperature of the air

Now that we have the enthalpy of the air after it has been heated, we can find the exit temperature using the specific heat of air: \(T_2 = \frac{h_2}{c_p} = \frac{792.75}{1.005} = 789.05 \mathrm{K}\) Since this temperature refers to the exiting air from the regenerator, this is our answer for part (a).

Calculate the mass flow rate of the exhaust gases

Using the same approach as in step 1, we can calculate the enthalpy of the exhaust gases entering and leaving the regenerator: \(h_3 = c_pT_3 = 1.005 \times 800 = 804 \mathrm{kJ/kg}\) \(h_4 = c_pT_4 = 1.005 \times 600 = 603 \mathrm{kJ/kg}\) Let's denote the mass flow rate of exhaust gases as \(\dot{m}_{exhaust}\). Using the energy balance equation and considering only heat input and output, we can write: \(\dot{m}_{air}(h_2 - h_1) = \dot{m}_{exhaust}(h_3 - h_4)\) Substituting in the given values and solving for the exhaust gas mass flow rate, we get: \(\dot{m}_{exhaust} = \frac{800\ (792.75 - 552.75)}{(804 - 603)} = \frac{800 \times 240}{201} = 954.23 \mathrm{kg/min}\) This is our answer for part (b), the mass flow rate of the exhaust gases.

Key concepts

Heat Transfer in Regenerative Heat Exchangers

Understanding how heat transfer works in regenerative heat exchangers is pivotal when it comes to increasing the efficiency of processes such as those used in gas-turbine power plants. In these systems, heat is recycled to pre-heat air entering the combustion chamber, saving energy and fuel. The equation for the rate of heat transfer, denoted as Q, is given by the product of the mass flow rate, \( \dot{m} \), and the change in specific enthalpy, \( \Delta h \). This can be observed in the calculation of the heat absorbed per unit mass by the air. In a regenerator, warm exhaust gases transfer energy to the cooler incoming air, increasing its enthalpy before it enters the combustion chamber. The effectiveness of this heat transfer process is essential for the plant's overall thermal efficiency.

Enthalpy and Its Role in Thermodynamics

Enthalpy, represented as \( h \), is a measurement of energy in a thermodynamic system and is particularly important when analyzing processes where pressure and temperature vary, such as in heat exchangers.

Enthalpy of Air

In the given exercise, we calculate the enthalpy of air before and after it passes through the regenerator by using the expression \( h = c_pT \), where \( c_p \) is the specific heat at constant pressure and \( T \) is the temperature. The increase in enthalpy, signifying the amount of heat absorbed by the air per kilogram, directly corresponds to the heat transferred from the exhaust gases to the air inside the regenerator.

Using the Ideal Gas Law in Calculations

The ideal gas law is a crucial factor in determining the state of a gas in thermal systems.

Applying Ideal Gas Law

In our context, it is used to calculate the specific volume of air entering the regenerator, with the formula \( pv = RT \), where \( p \) is the pressure, \( v \) is the specific volume, \( T \) is the temperature, and \( R \) is the specific gas constant. By rearranging the equation to \( v = \frac{RT}{p} \), we find the specific volume given the initial conditions of pressure and temperature. This step is fundamental in further calculations of enthalpy, where the gas constant \( R \) plays a vital role.

Specific Heat and Temperature Change

Specific heat, \( c_p \), is the amount of heat required to raise the temperature of one kilogram of a substance by one Kelvin.

Significance of Specific Heat

It is a material-specific value that is key to understanding how substances absorb and release heat.

• For air, \( c_p \), remains roughly constant at 1.005 \( \mathrm{kJ/kgK} \) within typical temperature ranges found in gas-turbine plants.
• In solving the regenerator problem, the specific heat allows us to convert the increase in enthalpy into a corresponding increase in temperature.

This concept helps us understand the energy changes within the air as it gains heat from the exhaust gases.

Energy Balance in Thermal Systems

The energy balance concept is the backbone of thermodynamics and process engineering, ensuring that all energy entering and exiting a system is accounted for.

Real-World Application

An energy balance equation, typically in the form of \( \dot{m}_{in}(h_{in}) = \dot{m}_{out}(h_{out}) \) when considering mass flows and enthalpies, allows engineers to calculate unknown variables such as mass flow rate or temperature in devices like regenerators. When the energy added to the air equals the energy removed from the exhaust gases, we achieve balance. This principle is central to determining the efficient operation of the regenerator and the entire power plant system.