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Chapter 8: Problem 18

A multistage compressor develops a total pressure ratio \(\pi_{c}=25\), and is designed with eight identical (i.e., "repeated") stages. The compressor polytropic efficiency is \(e_{\mathrm{c}}=0.92\). Calculate (a) average stage total pressure ratio \(\pi_{\mathrm{s}}\) (b) stage adiabatic efficiency \(\eta_{\mathrm{s}}\) (c) compressor total temperature ratio \(\tau_{c}\)

Short Answer

Expert verified
Numerical answers to (a), (b), and (c) based on the results from each step. For example: (a) \(\pi_{s} = \text{insert numerical solution here (a)}\), (b) \(\eta_{s} = \text{insert numerical solution here (b)}\), (c) \(\tau_{c} = \text{insert numerical solution here (c)}\)

Step by step solution

01

Calculate average stage total pressure ratio \(\pi_{s}\)

Given that we have eight identical stages and a total pressure ratio \(\pi_{c}=25\), we can determine the average stage total pressure ratio by taking the \(n\)th root of the total pressure ratio, where \(n\) is the stage number. This is calculated using the formula \(\pi_{s}=\sqrt[n]{\pi_{c}}\). Substituting \(n = 8\) and \(\pi_{c} = 25\) we find \(\pi_{s}\)
02

Calculate stage adiabatic efficiency \(\eta_{s}\)

The compressor polytropic efficiency \(e_{c} = 0.92\) is given. We know that the polytropic efficiency is related to the adiabatic efficiency by the formula \(\eta_{s}=\frac{e_{c}}{1-(1/e_{c})(\gamma-1)/\gamma*(1-\pi_s^{(\gamma-1)/\gamma})}\). Substituting \(e_{c} = 0.92\), \(\gamma = 1.4\) and the calculated value for \(\pi_{s}\) from the previous step, we find \(\eta_{s}\)
03

Calculate compressor total temperature ratio \(\tau_{c}\)

We know that the total temperature ratio \(\tau_{c}\) is related to the total pressure ratio \(\pi_{c}\) and the stage adiabatic efficiency \(\eta_{s}\) by the formula \(\tau_{c}=\pi_c^{(\gamma-1)/\gamma*\eta_s}\). Substituting \(\pi_{c} = 25\), \(\gamma = 1.4\) and the calculated value for \(\eta_{s}\) from the previous step, we find \(\tau_{c}\)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Polytropic Efficiency in Multistage Compressors
The concept of polytropic efficiency is a critical aspect in the study of multistage compressors. It refers to the efficiency of a process where compression takes place in very small, infinitesimally small, stages, such that each stage can be considered nearly reversible. In multistage compressors, the polytropic efficiency serves as a measure of how effectively the compressor performs in converting the input energy into useful work while minimizing losses due to heat dissipation.
To calculate polytropic efficiency, we use the ratio of the incremental work done during the compression to the incremental volume change. This enables the determination of how much energy is effectively utilized during the process.
  • A high polytropic efficiency implies minimal energy loss, resulting in better performance of the compressor.
  • In this exercise, the polytropic efficiency of the compressor is given as 0.92, indicating a high efficiency level.
  • This value is critical in further calculations like determining stage adiabatic efficiency and total temperature ratios.
Understanding Adiabatic Efficiency
Adiabatic efficiency is another important factor when analyzing compressors, especially when it comes to multistage systems. It measures how close a compressor's performance is to the ideal adiabatic process where no heat is lost or gained from the surroundings during compression. The adiabatic efficiency is lower than the polytropic efficiency due to real-world inefficiencies, such as friction and heat losses that occur in practical compressors.
The formula linking adiabatic efficiency to polytropic efficiency considers these imperfections, allowing engineers and students to estimate real-life performance more accurately. In our exercise, we calculate stage adiabatic efficiency using the provided polytropic efficiency, which requires understanding of the adiabatic process principles and thermodynamic properties of the gas (e.g., specific heat ratio, \(\gamma\)).
  • Adiabatic efficiency is crucial for understanding efficiency losses in compressor stages.
  • It is calculated using formulas that relate it to known polytropic efficiencies and pressure ratios.
  • The exercise requires substituting predefined values to obtain a reasonable estimate of adiabatic efficiency per stage.
Calculating Pressure Ratios
Calculating pressure ratios is foundational in analyzing compressor performance. The pressure ratio of a compressor is the ratio of the outlet pressure to the inlet pressure. In multistage compressors, each stage contributes to the overall pressure boost throughout the compressor.
In our scenario, the total pressure ratio \(\pi_{c} = 25\) is given for the compressor consisting of eight identical stages. To find the average stage total pressure ratio, \(\pi_{s}\), we use the formula for the nth root, \(\pi_{s}=\sqrt[n]{\pi_{c}}\), where \(n\) represents the number of stages. This calculation helps in understanding the load each compressor stage handles.
  • The total pressure ratio is a measure of the overall performance capability of the compressor.
  • The average stage pressure ratio enables engineers to design stages that handle expected pressure workloads effectively.
  • Correct calculations of stage pressures are vital for ensuring the compressor operates efficiently and predictably across all stages.

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Most popular questions from this chapter

The flow at the entrance to an axial-flow compressor rotor has zero preswirl and an axial velocity of \(175 \mathrm{~m} / \mathrm{s}\). The shaft angular speed is \(5000 \mathrm{rpm}\). If at a radius of \(0.5 \mathrm{~m}\), the rotor exit flow has zero relative swirl, calculate at this radius (a) rotor specific work \(w_{c}\) in \(\mathrm{kJ} / \mathrm{kg}\) (b) degree of reaction \({ }^{\circ} R\)

An axial flow compressor stage at \(r=0.4 \mathrm{~m}\) is shown. The inlet flow is purely axial with \(C_{21}=150 \mathrm{~m} / \mathrm{s}\), which remains constant across the rotor and stator. The rotor solidity at this radius is \(\sigma_{\mathrm{r}}=1.5\). The rotor angular speed is \(\omega=7400 \mathrm{rpm}\). The stage degree of reaction at this radius is \({ }^{\circ} R=0.75 .\) The rotor total pressure loss parameter (in relative frame) is \(\varpi_{r}=0.05 .\) The gas properties are: \(\gamma=1.4\) and \(R=287 \mathrm{~J} / \mathrm{kgK}\). Calculate: (a) absolute swirl at rotor exit, \(C_{\theta 2}\) in \(\mathrm{m} / \mathrm{s}\) (b) rotor exit (absolute) total temperature, \(T_{\mathrm{t} 2}\) in \(\mathrm{K}\) (c) static temperature at the rotor exit, \(T_{2}\) in \(\mathrm{K}\) (d) rotor exit absolute Mach number, \(M_{2}\) (e) rotor exit relative Mach number, \(M_{2 r}\) (f) relative dynamic pressure at rotor inlet, \(q_{1 r}\), in \(\mathrm{kPa}\) (g) inlet relative total pressure, \(p_{\mathrm{tIr}}\), in \(\mathrm{kPa}\) (h) static pressure at rotor exit, \(p_{2}\), in \(\mathrm{kPa}\) (i) rotor exit (absolute) total pressure, \(p_{12}\), in \(\mathrm{kPa}\) (j) rotor D-factor, \(D_{r}\), at this radius

An axial-flow compressor stage has a pitchline radius of \(r_{\mathrm{m}}=0.6 \mathrm{~m}\). The rotational speed of the rotor at pitchline is \(U_{\mathrm{m}}=256 \mathrm{~m} / \mathrm{s}\). The absolute inlet flow to the rotor is described by \(C_{z \mathrm{~m}}=155 \mathrm{~m} / \mathrm{s}\) and \(C_{\theta 1 \mathrm{~m}}=28 \mathrm{~m} / \mathrm{s}\). Assuming that the stage degree of reaction at pitchline is \({ }^{\circ} R_{\mathrm{w}}=0.50, \alpha_{3}=\alpha_{1}\), and \(C_{z \mathrm{~m}}\) remains constant, calculate (a) rotor angular speed \(\omega\) in rpm (b) rotor exit swirl \(\mathrm{C}_{\theta 2 \mathrm{~m}}\) (c) rotor specific work at pitchline, \(w_{c m}\) (d) relative velocity vector at the rotor exit (e) rotor and stator torques per unit mass flow rate (f) stage loading parameter at pitchline, \(\psi_{\mathrm{m}}\) (g) flow coefficient \(\varphi_{\mathrm{m}}\)

An axial-flow compressor stage is designed on the principle of constant through flow speed. The flow at the entrance to the rotor has \(100 \mathrm{~m} / \mathrm{s}\) of positive swirl and 180 \(\mathrm{m} / \mathrm{s}\) of axial velocity. Assuming we are at the pitchline radius \(r_{\mathrm{m}}=0.5 \mathrm{~m}\), where the rotor rotational speed is \(U_{\mathrm{m}}=230 \mathrm{~m} / \mathrm{s}\), the degree of reaction \({ }^{\circ} R_{\mathrm{m}}=0.5\), the radial shift in the streamtube is negligible, i.e., \(r_{1 \mathrm{~m}} \approx r_{2 \mathrm{~m}} \approx r_{3 \mathrm{~m}}\), and also assuming a repeated stage design principle is implemented, calculate (a) \(\alpha_{1 \mathrm{~m}}\) and \(\beta_{1 \mathrm{~m}}\) (b) \(\alpha_{2 m}\) and \(\beta_{2 m}\) (c) rotor specific work at the pitchline \(w_{\mathrm{cm}}\) in \(\mathrm{kJ} / \mathrm{kg}\) (d) stator torque at the pitchline per unit mass flow rate, \(\tau_{\mathrm{s}} / m\)

A compressor stage at the pitchline, \(r_{\mathrm{m}}=0.5 \mathrm{~m}\), is shown. The inlet flow to the rotor has a preswirl with \(\alpha_{1}=\) \(22^{\circ} .\) The axial velocity is \(C_{21}=C_{22}=C_{23}=170 \mathrm{~m} / \mathrm{s}\), i.e. constant throughout the stage. The stage is of repeated design, with \(\alpha_{3}=\alpha_{1} .\) Rotor and stator solidities at pitchline are 1.2 and \(1.0\) respectively. The rotor inlet relative velocity at pitchline is sonic, i.e., \(W_{1}=a_{1}\) and the rotor relative exit velocity, following de Haller criterion, is \(W_{2}=0.75 W_{1}\). Calculate: (a) shaft rotational speed, \(\omega\), in rpm (b) rotor exit (absolute) swirl, \(C_{\theta 2}\) in \(\mathrm{m} / \mathrm{s}\) (c) stage degree of reaction, \({ }^{\circ} R_{\mathrm{w}}\) (d) rotor D-factor, \(D_{\mathrm{r}}\) (e) stage total temperature ratio (f) stage total pressure ratio for \(e_{c}=0.9\)
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