Question

Develop an expression for the thermal efficiency of an ideal Brayton cycle with an ideal regenerator of effectiveness 100 percent. Use constant specific heats at room temperature.

Short answer

Question: Calculate the thermal efficiency of the ideal Brayton cycle with an ideal regenerator of 100% effectiveness, considering constant specific heats at room temperature. Answer: The thermal efficiency of an ideal Brayton cycle with an ideal regenerator of 100% effectiveness, considering constant specific heats at room temperature can be determined using the expression: η_thermal = [(T4 - T5) - (T2 - T1)] / (T4 - T3) where T1: Inlet temperature to the compressor T2: Exit temperature from the compressor, inlet temperature to the regenerator T3: Exit temperature from the regenerator, inlet temperature to the combustion chamber T4: Exit temperature from the combustion chamber, inlet temperature to the turbine T5: Exit temperature from the turbine, inlet temperature to the regenerator T6: Exit temperature from the regenerator (final state)

Step-by-step solution

Compression work

In an isentropic compression process (1-2), we use the relation: \(T_2 = T_1 \cdot (P_2/P_1)^{(\gamma-1)/\gamma}\). \(p_2 = p_3\) and \(p_1 = p_6\) The specific work done during compression \(w_c = c_p (T_2 - T_1)\).

Expansion work

In an isentropic expansion process (4-5), we use the relation: \(T_5 = T_4 \cdot (P_5/P_4)^{(\gamma-1)/\gamma}\). \(p_4 = p_3\) and \(p_5 = p_1\) The specific work done during expansion \(w_t = c_p (T_4 - T_5)\).

Heat addition

In the constant pressure heat addition process (3-4), the heat added \(q_{in} = c_p (T_4 - T_3)\).

Heat rejection

In the ideal regenerator, we assume 100% effectiveness. This means that all heat from the exhaust gases (turbine outlet) is transferred to the compressed air (compressor outlet). Thus, the heat rejected in the regenerator, \(q_{regen} = c_p (T_5 - T_6) = c_p (T_5 - T_2)\).

Net work done

The net work done (per unit mass) in the cycle is the difference between the turbine work and the compressor work, so \(w_{net} = w_t - w_c = c_p (T_4 - T_5) - c_p (T_2 - T_1)\).

Thermal efficiency

The thermal efficiency is given by the ratio of the net work done to the heat added, so: \(\eta_{thermal} = \cfrac{w_{net}}{q_{in}} = \cfrac{c_p (T_4 - T_5) - c_p (T_2 - T_1)}{c_p (T_4 - T_3)} = \cfrac{(T_4 - T_5) - (T_2 - T_1)}{(T_4 - T_3)}\). This is the final expression for the thermal efficiency of an ideal Brayton cycle with an ideal regenerator of 100% effectiveness, considering constant specific heats at room temperature.

Key concepts

Isentropic Process

An isentropic process is idealized thermodynamic process that is both adiabatic and reversible. In context of the Brayton cycle, which is a model for gas turbines, we encounter this process during compression (when air is compressed in the compressor) and expansion (when the gas expands in the turbine). The property of being adiabatic means that there is no exchange of heat with the surroundings, and reversible indicates that no entropy is generated within the system. We describe this process mathematically using the relation:
\(T_2 = T_1 \cdot (P_2/P_1)^{(\gamma-1)/\gamma}\).
Here, \(T_1\) and \(T_2\) are the initial and final temperatures, and \(P_1\) and \(P_2\) are the initial and final pressures respectively. The variable \(\gamma\) is the ratio of specific heats (Cp/Cv). Understanding this concept is crucial as it lays the foundation for calculating the work done in both the compression and expansion phases of the Brayton cycle.

Specific Work

The concept of specific work is fundamental to analyzing the performance of thermodynamic cycles like the Brayton cycle. It measures the work done per unit mass of the working fluid. In isentropic processes of the Brayton cycle, we calculate the specific work during compression and expansion phases as follows:

• For compression (process 1-2): \(w_c = c_p (T_2 - T_1)\)
• For expansion (process 4-5): \(w_t = c_p (T_4 - T_5)\)

Here, \(c_p\) represents the specific heat of the gas at constant pressure, and \(T\) are the respective temperatures at different stages of the cycle. The net specific work of the cycle is simply the difference between the work output from the turbine and the work input to the compressor. This calculation is vital for determining the thermal efficiency of the cycle, indicating how effectively the engine converts the heat into useful work.

Regenerator Effectiveness

Regenerator effectiveness is a measure of how well a regenerator recovers waste heat from the exhaust of a gas turbine in the Brayton cycle. With 100 percent effectiveness, the regenerator perfectly transfers the exhaust heat to the compressed air stream. This pre-heating reduces the required energy input for reaching the combustion temperature and improves overall cycle efficiency. The formula for the heat rejected in an ideal regenerator with 100% effectiveness is
\(q_{regen} = c_p (T_5 - T_2)\).
This expression assumes all the heat from the turbine exhaust (T_5) is used to preheat the air exiting the compressor (T_2). In practice, no regenerator can reach 100 percent due to physical constraints, but understanding this idealized scenario helps in optimizing the regenerator design for improved thermal performance.

Constant Specific Heats

The assumption of constant specific heats simplifies calculations in the Brayton cycle. Specific heats, represented as \(c_p\) (constant pressure) and \(c_v\) (constant volume), typically vary with temperature. However, for many engineering applications involving gases, these variations are small within the temperature range of interest - often around room temperature. Therefore, we assume they remain constant which makes it easier to integrate them into thermodynamic equations, resulting in simplified expressions for various cycle parameters such as work and efficiency.
Using constant specific heats allows us to derive a concise formula for the thermal efficiency of an ideal Brayton cycle with an ideal regenerator. This approach is particularly useful for educational purposes, providing clear insights into the fundamental behaviors of the cycle without the complexity of temperature-dependent specific heats.