Question

Products of combustion enter a gas turbine with a stagnation pressure of \(0.75 \mathrm{MPa}\) and a stagnation temperature of \(690^{\circ} \mathrm{C}\), and they expand to a stagnation pressure of \(100 \mathrm{kPa}\) Taking \(k=1.33\) and \(R=0.287 \mathrm{kJ} / \mathrm{kg} \cdot \mathrm{K}\) for the products of combustion, and assuming the expansion process to be isentropic, determine the power output of the turbine per unit mass flow.

Short answer

Question: Determine the power output per unit mass flow of a gas turbine, given an isentropic expansion process with the following parameters: initial stagnation temperature \(T_1 = 690^{\circ}\mathrm{C}\), stagnation pressures \(P_1 = 0.75\,\text{MPa}\) and \(P_2 = 100\,\text{kPa}\), and gas properties \(k = 1.33\) and \(R = 0.287\,\mathrm{kJ/kg\cdot K}\). Solution: First, calculate the initial and final temperatures using the isentropic relations, then determine the work done by the gas turbine using the temperature differences and specific gas constant. Finally, calculate the power output per unit mass flow using the given values and calculated temperatures. The power output per unit mass flow, \(w\), can be found by evaluating the expression \(w = 0.287\,\mathrm{kJ/kg\cdot K}\left(963.15\,\mathrm{K} - T_2\right)\) after calculating \(T_2\).

Step-by-step solution

Calculate the Initial and Final Temperatures

First, we need to convert the given stagnation temperatures to Kelvin. The initial stagnation temperature is \(T_1 = 690^{\circ}\mathrm{C}+273.15=963.15\mathrm{K}\). To find the final temperature \(T_2\), we can use the isentropic relation for stagnation temperatures and pressures: \(\frac{T_{2}}{T_{1}}=\left(\frac{P_{2}}{P_{1}}\right)^{\frac{k-1}{k}}\) Given stagnation pressures, \(P_1=0.75\,\text{MPa}=750\,\text{kPa}\) and \(P_2=100\,\text{kPa}\) and \(k=1.33\), we can solve for \(T_2\): \(T_2 = T_1 \left(\frac{P_2}{P_1}\right)^{\frac{k-1}{k}}\)

Determine the Work Done

Since the expansion process is isentropic, the work done (\(w\)) by the gas turbine can be determined using specific gas constant \(R\) and the temperature difference between the initial and final states: \(w = R(T_1-T_2)\)

Calculate the Power Output per Unit Mass Flow

Now we can plug in the given values and the calculated final temperature to determine the power output of the turbine per unit mass flow: \(w = R(T_1-T_2) = 0.287\,\mathrm{kJ/kg\cdot K}\left(963.15\,\mathrm{K} - T_2\right)\) After calculating \(T_2\) in step 1 and substituting it into the equation, we can then solve for \(w\), which will be our final answer for the power output per unit mass flow.