Question

A gas turbine for an automobile is designed with a regenerator. Air enters the compressor of this engine at \(100 \mathrm{kPa}\) and \(30^{\circ} \mathrm{C}\). The compressor pressure ratio is \(10 ;\) the maximum cycle temperature is \(800^{\circ} \mathrm{C} ;\) and the cold air stream leaves the regenerator \(10^{\circ} \mathrm{C}\) cooler than the hot air stream at the inlet of the regenerator. Assuming both the compressor and the turbine to be isentropic, determine the rates of heat addition and rejection for this cycle when it produces 115 kW. Use constant specific heats at room temperature.

Short answer

Question: Determine the rates of heat addition and rejection for a gas turbine engine with a regenerator, given the following information: Inlet air conditions: pressure (\(P_1 = 100\,\mathrm{kPa}\)), temperature (\(T_1 = 30^{\circ}\mathrm{C}\)); compressor pressure ratio (\(r_c = 10\)); maximum cycle temperature (\(T_3 = 800^{\circ}\mathrm{C}\)); and temperature difference between the hot and cold air streams in the regenerator (\(\Delta T_\mathrm{reg}\)). Assume that both the compressor and the turbine are isentropic and use constant specific heats at room temperature. Solution: After performing the step-by-step calculation explained above, we can determine the rates of heat addition and rejection for the gas turbine engine with a regenerator.

Step-by-step solution

Path of the Problem

To solve this exercise, the main steps are: 1. Find the air temperatures at different stages of the cycle using isentropic relations 2. Calculate the work-input to the compressor and work-output from the turbine using energy balance equations 3. Then, calculate the net work output, based on the information of the engine producing 115 kW 4. Compute the rates of heat addition and rejection in the cycle

Calculate air temperatures

We must first calculate the air temperatures at different stages of the cycle, \(T_2\), \(T_4\) and \(T_5\), following isentropic relations: For isentropic processes, \(T_{2}/T_{1} = (P_{2}/P_{1})^{(\gamma-1)/\gamma}\), where \(\gamma\) is the ratio of specific heats, and \(P_{2} = r_c \cdot P_{1}\): \(T_2 = T_1 \cdot (r_c)^{(\gamma-1)/\gamma}\) The temperature \(T_4\) can be obtained from the energy balance equation: \(T_4 = T_3 - (T_2 - T_1)\) Finally, the temperature \(T_5\) is found as: \(T_5 = T_4 - \Delta T_\mathrm{reg}\)

Work-input to compressor and work-output from turbine

Next, we calculate the specific work-input to the compressor and work-output from the turbine using energy balance equations and the constant specific heat values at room temperature (\(c_p\) and \(c_v\)): Specific work-input to the compressor: \(w_\mathrm{in} = c_p \cdot (T_2 - T_1)\) Specific work-output from the turbine: \(w_\mathrm{out} = c_p \cdot (T_3 - T_4)\)

Calculate the mass flow rate

Now, we can find the net work output, \(W_\mathrm{net} = w_\mathrm{out} - w_\mathrm{in}\). Given that the engine produces 115 kW, the mass flow rate can be calculated as: \(\dot{m} = \dfrac{W_\mathrm{net}}{w_\mathrm{out} - w_\mathrm{in}}\)

Calculate the rates of heat addition and rejection

Finally, using the mass flow rate, we can compute the rates of heat addition and rejection in the cycle: Heat addition rate: \(\dot{Q}_\mathrm{in} = \dot{m} \cdot c_p \cdot (T_3 - T_2)\) Heat rejection rate: \(\dot{Q}_\mathrm{out} = \dot{m} \cdot c_v \cdot (T_4 - T_5)\) Calculating these values with the given input data and assumptions will give the desired rates of heat addition and rejection for the gas turbine engine with a regenerator.