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Chapter 9: Problem 2

Size the exit radius of a centrifugal compressor impeller, \(r_{2}\), that is to reach a tangential Mach number of \(M_{\mathrm{T}}=1.5\). The shaft rotational speed is \(25,000 \mathrm{rpm}\) and the inlet flow condition is characterized by $$ \begin{aligned} p_{1} &=100 \mathrm{kPa} \\ T_{1} &=288 \mathrm{~K} \\ M_{1} &=0.5 \\ \gamma &=1.4 \text { and } c_{p}=1,004 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K} \end{aligned} $$

Short Answer

Expert verified
The exit radius, \(r_2\), is calculated by sequentially determining the tangential velocity at the inlet and exit using given and calculated parameters. Finally, the exit radius is found by dividing the obtained tangential exit velocity by the rotational speed of the shaft.

Step by step solution

01

Calculating the Tangential Velocity at the Inlet

The tangential velocity at the inlet, \(c_1\), can be calculated using the formula: \(c_{1}=M_{1} \sqrt{\gamma R T_{1}}\) where \(R=c_{p}/(\gamma鈭�1)\).
02

Calculating the Tangential Velocity at the Exit

The tangential velocity at the exit, \(c_2\), can be calculated using the formula: \(c_{2}=M_{t} \sqrt{\gamma R T_{1}}\) where \(M_t\) is the given Mach number at the exit.
03

Calculating the Shaft Rotational Speed in rad/s

The shaft rotational speed, 蠅, in rad/s can be calculated from the provided RPM value using the formula: \(蠅 = RPM \times \frac{2蟺}{60}\)
04

Calculating the Exit Radius

Lastly, calculate the impeller exit radius. The exit radius, \(r_2\), can be found using the formula: \(r_{2}=\frac{c_2}{蠅}\)

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Key Concepts

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

Tangential Mach Number
In centrifugal compressor design, understanding the tangential Mach number, \( M_T \), is crucial, because it relates the speed of the fluid moving tangentially to the impeller to the speed of sound within the fluid. The tangential Mach number is a way to express compressibility effects and potential shock waves formation within the flow. When the tangential Mach number approaches or exceeds one, compressibility effects become significant, which can impact efficiency and performance. Therefore, knowing and controlling \( M_T \) helps ensure the compressor operates efficiently. In our example, reaching a tangential Mach number of 1.5 implies that the fluid is moving at 1.5 times the speed of sound tangentially at the exit of the impeller. It highlights the importance of aerodynamic design in ensuring that supersonic conditions do not lead to shock losses.
Shaft Rotational Speed
The shaft rotational speed is a fundamental parameter in the design and operation of centrifugal compressors. It measures how quickly the compressor shaft rotates, typically expressed in revolutions per minute (RPM). Understanding this speed is essential as it directly influences the kinetic energy imparted to the fluid. To convert RPM into a more usable format for calculations, we express it in radians per second, \( \omega \), using the formula \( \omega = \text{RPM} \times \frac{2\pi}{60} \).

For our example, the shaft rotational speed is given as 25,000 RPM, which translates to a rotational speed in rad/s after applying the conversion. This value is then used in further calculations to determine critical aspects like the impeller exit radius. A meticulously established shaft speed ensures that the compressor operates within desired parameters, influencing efficiency and mechanical integrity.
Impeller Exit Radius
The impeller exit radius \( r_2 \) is a key geometric parameter in centrifugal compressors. It represents the distance from the shaft center to the outer edge of the impeller where the air exits. This radius is vital since it dictates the available space for the fluid to exit, affecting pressure and the efficiency of the compressor.
  • To compute \( r_2 \), we first determine the tangential velocity at the exit \( c_2 \) using the tangential Mach number.
  • With this velocity and the shaft rotational speed \( \omega \) obtained earlier, we apply the formula: \( r_2 = \frac{c_2}{\omega} \).
This calculation is crucial as it confirms that the designed exit radius will properly accommodate the specified Exit Mach number and rotational velocity, ensuring optimal compressor function.
Inlet Flow Conditions
Inlet flow conditions describe the thermodynamic state of the fluid before it enters the impeller of a centrifugal compressor. Understanding these conditions is vital for predicting and optimizing the performance of the compressor. Key parameters include pressure \( p_1 \), temperature \( T_1 \), and inlet Mach number \( M_1 \).
  • Pressure \( p_1 \) affects air density and energy transfer capability, given here as 100 kPa.
  • The temperature \( T_1 \), at 288 K, relates to air density and speed of sound, impacting mach numbers and resulting computations.
  • The inlet Mach number \( M_1 = 0.5 \), indicates the flow velocity relative to the speed of sound at the inlet.
These inlet conditions are used to calculate the tangential velocity \( c_1 \) which, along with the rotational speed, helps in evaluating the compressor performance and ensuring it meets the required specifications.

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Most popular questions from this chapter

A centrifugal compressor impeller has 20 straight blades. The exit radius of the impeller is \(r_{2}=0.30 \mathrm{~m}\) and the angular speed is \(\omega=14,500 \mathrm{rpm}\). The inlet flow to the compressor has zero preswirl and the axial velocity is \(C_{21}=\) \(150 \mathrm{~m} / \mathrm{s}\). The impeller exit has the radial velocity component \(C_{n 2}\) that is equal to \(C_{21}\). Assuming the speed of sound at the inlet to compressor is \(a_{1}=300 \mathrm{~m} / \mathrm{s}\), calculate (a) Mach index, \(\Pi_{\mathrm{M}}\) (b) impeller exit absolute swirl, \(C_{\theta 2}\), in \(\mathrm{m} / \mathrm{s}\) (c) specific work of the rotor, w \(_{c}\), in \(\mathrm{kJ} / \mathrm{kg}\) (d) impeller exit absolute Mach number, \(M_{2}\) (e) compressor total temperature ratio, \(\tau_{c}\)

The corrected mass flow rate at the inlet to a centrifugal compressor is \(\dot{m}_{c 1}=20 \frac{k s}{x}\) and the axial Mach number at the compressor face is \(M_{21}=0.5\). For the hub-to-tip radius ratio of the impeller equal to \(0.1\), i.e., \(r_{\mathrm{h} 1} / r_{11}=0.1\), calculate (a) inlet area, \(A_{1}\), in \(m^{2}\) (b) hub and tip radii, \(r_{\mathrm{al}}\) and \(r_{\mathrm{t} 1}\), in \(\mathrm{cm}\) (c) shaft speed, \(\omega\), for the impeller inlet relative tip Mach number to be \(0.8\), i.c., \(\left(M_{1 \mathrm{r}}\right)_{\text {lip }}\) Assume gas total temperature at the impeller inlet is \(288 \mathrm{~K}\), \(\gamma=1.4\) and \(R=287 \mathrm{~J} / \mathrm{kgK}\)

The inlet static pressure to an impeller is \(p_{1}=100 \mathrm{kPa}\) and its exit static pressure is \(p_{2}=200 \mathrm{kPa}\) with inlet relative dynamic pressure \(q_{1 \mathrm{t}}=140 \mathrm{kPa}\). Treating the impeller as a diffuser, assuming \(\gamma=1.4\), calculate (a) inlet relative Mach number \(M_{\mathrm{lr}}\) (b) static pressure recovery coefficient \(C_{\mathrm{PR}}\)

A centrifugal compressor has 25 impeller blades of radial design. The rotor exit diameter is \(r_{2}=0.4 \mathrm{~m}\), and its rotational speed is \(\omega=8000 \mathrm{rpm}\). Assuming the inlet to rotor flow is purely axial and the air mass flow rate is \(\dot{m}=25 \mathrm{~kg} / \mathrm{s}\) through the compressor, calculate (a) compressor shaft power, \(8_{c}\) in, \(\mathrm{MW}\) (b) (time rate of change of) angular momentum at the rotor exit

A centrifugal compressor impeller is of radial design and has 18 blades. The impeller exit radius is at \(r_{2}=0.30 \mathrm{~m}\). The wheel rim speed is \(U_{2}=400 \mathrm{~m} / \mathrm{s}\). Assuming that flow in compressor inlet is swirl frec, the axial velocity at the inlet and the radial velocity at the exit of the impeller are equal, i.e., \(C_{21}=C_{22}=150 \mathrm{~m} / \mathrm{s}\), with \(T_{\mathrm{n}}=288 \mathrm{~K}, p_{\mathrm{ul}}=100 \mathrm{kPa}\), \(\gamma=1.4\) and \(c_{p}=1004 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\), calculate (a) absolute swirl at the impeller exit, \(C_{\theta 2}\), in \(\mathrm{m} / \mathrm{s}\) (b) absolute Mach number at the impeller exit, \(M_{2}\) (c) absolute radial and tangential velocitics at the radial diffuser exit, \(C_{83}\) and \(C_{83}\), in \(\mathrm{m} / \mathrm{s}\) (Assume the fluid is incompressible and inviscid in the radial diffuser)
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