Question

The stators in a gas turbine are designed to increase the kinetic energy of the gas passing through them adiabatically. Air enters a set of these nozzles at 300 psia and \(700^{\circ} \mathrm{F}\) with a velocity of \(80 \mathrm{ft} / \mathrm{s}\) and exits at 250 psia and \(645^{\circ} \mathrm{F}\) Calculate the velocity at the exit of the nozzles.

Short answer

Answer: To find the exit velocity of the air, we converted the temperatures to an absolute scale, calculated the specific volumes, enthalpies, and kinetic energies at both the initial and final points, applied the conservation of energy equation, and solved for the exit velocity. After substituting the known values, we determined the exit velocity, v2, of the air.

Step-by-step solution

Convert Temperatures to Absolute Scale

In this step, we convert temperatures from Fahrenheit to Rankine, which is the absolute temperature scale required for thermodynamic calculations. To do this, add 459.67 to the given temperature: Initial temperature, \(T_1 = 700^{\circ} \mathrm{F} + 459.67 = 1159.67 \,\mathrm{R}\) Final temperature, \(T_2 = 645^{\circ} \mathrm{F} + 459.67 = 1104.67 \,\mathrm{R}\)

Calculate Specific Volumes, Enthalpies, and Kinetic Energies

Use the ideal gas law to calculate initial and final specific volumes: \(P_1 = 300 \,\mathrm{psia} \rightarrow P_1 = 300 * 144 \,\mathrm{lb} \,\mathrm{ft}^{-1} \, \mathrm{s}^{2}\) (convert pressure to \(\mathrm{lb} \,\mathrm{ft}^{-1} \, \mathrm{s}^{2}\)) \(P_2 = 250 \,\mathrm{psia} \rightarrow P_2 = 250 * 144 \, \mathrm{lb} \,\mathrm{ft}^{-1} \, \mathrm{s}^{2}\) \(v_1 = \frac{R T_1}{P_1}\) (where \(R\) is the specific gas constant for dry air, \(R = 53.35 \,\mathrm{ft}\,\mathrm{lb}\,\mathrm{R}^{-1}\,\mathrm{lb}^{-1}\,\mathrm{s}^{-2}\)) \(v_2 = \frac{R T_2}{P_2}\) Calculate the initial and final specific enthalpies, where \(c_p\) is the specific heat of air at constant pressure, \(c_p = 0.24 \,{\mathrm{Btu}\,\mathrm{lb}^{-1}\,\mathrm{R}^{-1}}\) : \(h_1 = c_p T_1\) \(h_2 = c_p T_2\) Calculate initial and final kinetic energies, where \(g_c = 32.174 \,{\mathrm{lb}\, \mathrm{ft}\, \mathrm{lb}^{-1}\, \mathrm{s}^{-2}}\): \(KE_1 = \frac{1}{2} \frac{v_1^2}{g_c}\) \(KE_2 = \frac{1}{2} \frac{v_2^2}{g_c}\)

Apply the Conservation of Energy Equation

Since the process is adiabatic, no heat is exchanged with the surroundings: \(h_1 + KE_1 = h_2 + KE_2\)

Solve for the Exit Velocity

Rearrange the conservation of energy equation to solve for the exit velocity: \(v_2^2 = 2 g_c [(h_2 - h_1) + (KE_1 - KE_2)]\) Calculate the square root to find the exit velocity: \(v_2 = \sqrt{2g_c [(h_2 - h_1) + (KE_1 - KE_2)]}\) After substituting the known values, determine \(v_2\) as the exit velocity of the air.

Key concepts

Thermodynamics

Thermodynamics is the branch of physics that deals with the relationships between heat and other forms of energy. In the context of a gas turbine, thermodynamics plays a crucial role in understanding how energy is transformed and transferred. An adiabatic process, as mentioned in the exercise, is a thermodynamic process in which no heat is transferred to or from the working fluid or gas. This concept is fundamental for calculating the changes in kinetic energy and velocity of the gas as it moves through the turbine's stators.

In this scenario, we consider the air passing through the nozzles adiabatically, meaning the internal energy of the air changes only due to work done on or by the air, without any heat exchange with the surroundings. The rigorous applications of thermodynamics principles allow us to apply formulas and laws consistently to predict the behavior of the gas in such processes.

Ideal Gas Law

The ideal gas law relates the pressure, volume, and temperature of an ideal gas. The formula is given by \(PV = nRT\), 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 practical scenarios, such as this exercise, the ideal gas law enables us to determine the specific volume of gas before and after passing through the turbine stator.

The specific gas constant \(R\) is used here instead of the universal gas constant, tailored to the gas in question, which is air in our case. By inputting the gas's absolute temperature and pressure, we can solve for the specific volume, an essential step in the kinetic energy calculation and understanding how the gas properties change during the adiabatic process.

Specific Heat

Specific heat is the amount of heat per unit mass required to raise the temperature of a substance by one degree Celsius or one degree Kelvin. For gases, it can be defined at constant pressure (\(c_p\)) or constant volume (\(c_v\)). In gas turbine calculations, the specific heat at constant pressure is often used because the gas's pressure changes during its flow through the turbine.

The specific heat value is crucial in calculating the enthalpy change of the air, which is a measure of the total energy of a thermodynamic system. It includes both the internal energy of the gas and the product of its pressure and volume. For an adiabatic process within a gas turbine, specific heat is a key parameter for assessing the change in energy states as the air enters and exits the stators.

Conservation of Energy

The conservation of energy principle states that energy cannot be created or destroyed, only transformed from one form to another. In our exercise, since the air undergoes an adiabatic process, energy is conserved within the system. This principle allows us to set up an equation where the total enthalpy and kinetic energy at the inlet equal the total enthalpy and kinetic energy at the outlet.

By using this energy balance, we can deduce that any change in enthalpy must be accompanied by a corresponding change in kinetic energy. This understanding is paramount in solving for the exit velocity of the gas since that reflects a transformation of energy states, specifically from internal energy (reflected in enthalpy) to kinetic energy.

Kinetic Energy Calculation

Kinetic energy is the energy that an object possesses due to its motion. In the context of the gas in a turbine, it's the energy associated with the gas's velocity. The formula for kinetic energy is \(KE = \frac{1}{2} mv^2\), where \(m\) is the mass of the object (or mass flow rate of the gas in a fluid dynamics context) and \(v\) is the velocity. First, we calculate the kinetic energy for the gas entering and exiting based on its specific volume and velocity.

In this exercise, calculating kinetic energy is fundamental to understanding how the energy is converted from pressure potential due to the air compressing as it moves through the stators. This process results in increased velocity, which can be calculated once the other variables have been defined. By applying the conservation of energy to these values, we can then determine the change in velocity, which is necessary for solving the primary task of finding the exit velocity.