Question

Consider a simple ideal Brayton cycle operating between the temperature limits of 300 and 1500 K. Using constant specific heats at room temperature, determine the pressure ratio for which the compressor and the turbine exit temperatures of air are equal.

Short answer

Answer: The pressure ratio for which the compressor and turbine exit temperatures are equal is approximately 7.49.

Step-by-step solution

Review the Brayton cycle and properties

The ideal Brayton cycle consists of four processes: 1-2: Isentropic compression, 2-3: Constant pressure heat addition, 3-4: Isentropic expansion, and 4-1: Constant pressure heat rejection. For an ideal gas with constant specific heat, we can write the relation between temperatures and pressures in isentropic compression and expansion processes. For process 1-2 (Compressor): T2/T1 = (P2/P1)^((k-1)/k) For process 3-4 (Turbine): T4/T3 = (P4/P3)^((k-1)/k) Given T1 = 300 K and T3 = 1500 K, and since P2 = P3 and P1 = P4, let's denote the pressure ratio as r = P2/P1 = P3/P4.

Determine the exit temperatures and equate them

Now, we need to determine the compressor exit temperature T2 and the turbine exit temperature T4. Using the pressure ratio r and the specific heat ratios, we can substitute the temperature ratios in the previous formulas to find them. T2 = T1 * (r)^((k-1)/k) T4 = T3 * (1/r)^((k-1)/k) Now, we have to find the pressure ratio r in a way that the exit temperatures T2 and T4 are equal: T2 = T4 T1 * (r)^((k-1)/k) = T3 * (1/r)^((k-1)/k)

Solve for the pressure ratio

Solving for r, we get: r^((2(k-1))/k) = T3/T1 To solve for r, we can first find the value of (2(k-1))/k for an air, specific heat ratio (k) is 1.4: r^((2(1.4-1))/1.4) = 1500/300 Solving for r: r = (1500/300)^(1.4/0.8) r ≈ 7.49 Therefore, the pressure ratio for which the compressor and turbine exit temperatures are equal is approximately 7.49.

Key concepts

Isentropic Compression

The term isentropic compression is vital when describing one of the stages in the Brayton cycle. The concept refers to the compression of a gas without an increase or decrease in entropy, which is a measure of disorder or randomness in a system. During this process, energy is applied to compress the gas, leading to an increase in its pressure and temperature while maintaining a consistent entropy value.

For gases behaving ideally, and with specific heats assumed constant, the temperature and pressure relationship during isentropic compression follows a precise mathematical relationship. Mathematically, this relationship can be expressed using the equation:
\[ \frac{T2}{T1} = \left(\frac{P2}{P1}\right)^{\left(\frac{k-1}{k}\right)} \]
where \( T1 \) and \( T2 \) are the initial and final temperatures, \( P1 \) and \( P2 \) are the initial and final pressures, and \( k \) is the specific heat ratio. Understanding this process is crucial for engineers and students as it greatly influences the efficiency and output of gaseous cycles like the Brayton cycle used in power plants and jet engines.

For example, if a gas within a compressor is initially at a temperature of 300 K, and it undergoes isentropic compression to a higher pressure, we can determine the new temperature at this pressure using the above equation. Optimizing this stage ensures the overall efficiency of the Brayton cycle, where too much compression can lead to temperature increases that may degrade engine components over time.

Pressure Ratio

The pressure ratio in the Brayton cycle is a critical factor affecting the overall performance of the system. It is defined as the ratio of the pressure after the compression process to the pressure before the compression process. Represented mathematically by \( r = \frac{P2}{P1} \), the pressure ratio directly correlates with the temperature increase during the isentropic compression phase.

The ideal pressure ratio is significant because it determines the cycle's thermal efficiency and the temperatures at the exits of the compressor and turbine. To ensure that the compressor and turbine exit temperatures are equal, which is often a design consideration for efficiency and simplicity in some systems, we must carefully calculate the optimal pressure ratio.

In solving for the pressure ratio, we equate the temperature at the exit of the compressor and the turbine, resulting in an equation from which the pressure ratio can be deduced. For example, with the given temperatures of 300 K and 1500 K for the air intake and after the combustion stage, respectively, and assuming air follows the ideal gas model with constant specific heat, this ratio is essential to balance the cycle's input and output conditions.

As we observed in the provided step-by-step solution, when the Brayton cycle's entry and exit temperatures are known, the pressure ratio can be manipulated to ensure optimal efficiency and operational stability, ultimately leading to the Brayton cycle's desired performance.

Specific Heat

The concept of specific heat plays a critical role in understanding thermodynamic cycles such as the Brayton cycle. Specific heat is the amount of heat required to change the temperature of a unit mass of a substance by one-degree Celsius. It is an intrinsic property of the material and, for an ideal gas, it is assumed to remain constant with temperature changes within certain temperature ranges.

In the Brayton cycle, the specific heat at constant pressure \( (c_p) \) and the specific heat at constant volume \( (c_v) \) are used to define the specific heat ratio \( (k) \), where \( k = \frac{c_p}{c_v} \). The value of \( k \) influences the isentropic compression and expansion processes, as seen in the formula:
\[ T2 = T1 \times r^{\left(\frac{k-1}{k}\right)} \]
where \( r \) is the pressure ratio.

For an ideal gas like air, the specific heat ratio is approximately 1.4 under normal conditions. This ratio is vital in calculating the pressure ratio necessary to equalize the compressor and turbine exit temperatures in the Brayton cycle, ensuring the cycle is optimized for the desired application. Moreover, the assumption of constant specific heats simplifies the equations and calculations involved in thermodynamic analyses, although more advanced models may require variable specific heat considerations for greater accuracy at different temperature ranges.

Understanding how specific heat relates to the workings of the Brayton cycle can help students and professionals appreciate the intricacies of energy conversion in gas turbines and jet engines where this cycle is predominantly used.