2. Centrifugal compressor and the basic steps involved in its design

The design of the centrifugal compressor is very difficult due to the reasons like it is a gasdriven turbo-machinery component, involvement of extensive iterative process for the convergence of the design, enormous design complexity due to three-dimensional flow phenomena, and multiflow physics embedded within a dynamic state-of-the-art. Hence, in this article, a detailed procedure for the design is presented based on the design methodology as explained in the literature [1–3]. The basic steps involved in the design of the centrifugal compressor are discussed in detail in the following sections.

### 2.1. Impeller and velocity triangles

Impeller is the main design task during the phase of the compressor design. There are various strategic geometric/design features to be identified and discussed with respect to impeller, its inlet and outlet design. In principal, aerodynamic losses occurring in majority of turbomachines arise primarily due to the boundary layer growth, its separation on blade profile, and passage surfaces termed as profile losses widely configured under primary losses.

The velocity triangles of the impeller of a centrifugal compressor are diagrammatically shown in Figure 1.

Figure 1. Velocity triangles of impeller for a centrifugal compressor.

The impeller design is the basic and major vital part of the design of the compressor. The following conditions for the design may be considered as limiting conditions


### 2.2. Specific work

efficiency, and reduced structural weight of compressor and the engine as well. Various compressor stages achieve gradual increase in the stagnation-to-flow pressure contributed by flow diffusion. Energy is added in the rotor blade section, increasing the total pressure and absolute component of flow velocity. Stator blade row diffuses the flow, thus reducing absolute velocity component and elevating static pressure. Blade topology requires adaptation of a cautious design procedure to achieve the designated pressure rise while minimizing aero-thermodynamic losses

In this chapter, a strong attempt is made to enumerate the detailed procedure of the centrifugal compressor in Section 2, concepts and basics of Numerical Schemes in Section 3. Further, a case study is discussed in detail in Section 4, the corresponding results and discussions are

The design of the centrifugal compressor is very difficult due to the reasons like it is a gasdriven turbo-machinery component, involvement of extensive iterative process for the convergence of the design, enormous design complexity due to three-dimensional flow phenomena, and multiflow physics embedded within a dynamic state-of-the-art. Hence, in this article, a detailed procedure for the design is presented based on the design methodology as explained in the literature [1–3]. The basic steps involved in the design of the centrifugal compressor are

Impeller is the main design task during the phase of the compressor design. There are various strategic geometric/design features to be identified and discussed with respect to impeller, its inlet and outlet design. In principal, aerodynamic losses occurring in majority of turbomachines arise primarily due to the boundary layer growth, its separation on blade profile,

The velocity triangles of the impeller of a centrifugal compressor are diagrammatically shown

and passage surfaces termed as profile losses widely configured under primary losses.

2. Centrifugal compressor and the basic steps involved in its design

in order to run and achieve design pressure ratios and design efficiencies.

presented in Section 5. Final conclusions are summarized in Section 6.

discussed in detail in the following sections.

Figure 1. Velocity triangles of impeller for a centrifugal compressor.

2.1. Impeller and velocity triangles

98 Numerical Simulations in Engineering and Science

in Figure 1.

The specific work can be calculated using the following correlation, which is based on energy and Euler turbine equations

$$(\mu\_2 \mathbf{C}\_{u,2} - \mu\_1 \mathbf{C}\_{u,1}) = \mathbf{g}\_c \mathbf{C}\_p \mathbf{g}\_c T\_{0,1} \left\{ \left( \frac{p\_{0,2}}{p\_{0,1}} \right)^{\left[ \left( \frac{p}{C\_p} \right)^{\frac{1}{\eta\_p c}} \right]} - 1 \right\} \tag{1}$$

### 2.3. Slip factor

In general for the design of the compressor, the inlet swirl is considered as zero and hence, Cu,<sup>1</sup> ¼ 0. Furthermore, the Wiesner's correlation for slip factor is given by

$$\sigma\_w = \text{slipfactor} = \frac{\mathbb{C}\_{u,2,ac}}{\mathbb{C}\_{u,2,dl}} = 1 - \frac{\sqrt{\cos \beta\_2}}{Z^{0.7}} \tag{2}$$

where number of blades is denoted by Z and β<sup>2</sup> represents the angle between the radial direction and tangent to the rotor blade at the periphery.

αc,<sup>2</sup> should be less than 60� such that the downstream diffuser does not get prone to stall when C2and Cr, <sup>2</sup> are reduced by keeping very large αc, 2.

### 2.4. Number of blades

The number of blades and the exit blade angle are dependent on each other and can be calculated using loading coefficient. Loading coefficient ψ is the ratio between the outlet tangential flow velocity and blade speed, which is given by the correlation

$$\psi = \frac{\mathbb{C}\_{u,2,\text{sc}}}{\mu\_2} = \left\{ \left[ \frac{\tan \beta\_2}{\tan \alpha\_{c,2}} \right] + \frac{1}{\sigma\_w} \right\}^{-1} \tag{3}$$

Cu, <sup>2</sup>, ac ¼ ψu<sup>2</sup> (8)

http://dx.doi.org/10.5772/intechopen.72098

(9)

101

(10)

(12)

Cr, <sup>2</sup> <sup>¼</sup> Cu,2, ac tan αc, <sup>2</sup>

Numerical Simulations of a High-Resolution RANS-FVDM Scheme for the Design of a Gas Turbine Centrifugal…

<sup>C</sup><sup>2</sup> <sup>¼</sup> Cu,2, ac sin αc, <sup>2</sup>

<sup>R</sup> <sup>1</sup> � <sup>1</sup> <sup>þ</sup>

4

2 4

M<sup>2</sup> 2 Cp <sup>R</sup> � 1 � �

> gcΔh<sup>0</sup> � �3=<sup>4</sup> ffiffiffiffi

The outlet blade axial width is calculated with the assumptions that the effect of the thickness and boundary layer of the blade are neglected. The following correlations are used for the

<sup>u</sup><sup>2</sup> <sup>¼</sup> <sup>π</sup>d2<sup>N</sup>

Flow separation should be minimized in the design. To reduce the separation of the flow, the back sweep angle should be higher. Furthermore, the separation in the rotor with subsonic inlet relative Mach number and normal Reynold's number can be minimized by using a lower

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

3 5

Cp <sup>R</sup> �1 � �

2 �<sup>1</sup>

M<sup>2</sup> 2 Cp <sup>R</sup> � 1 � �

3 5

3 5 vuuut (11)

<sup>V</sup>\_ <sup>p</sup> (13)

<sup>60</sup> (15)

m\_ ¼ Cr, <sup>2</sup>rst,2πd2b<sup>2</sup> (14)

The Mach number at the inlet can be calculated using the following correlation

2 Cp

¼ 1 þ

<sup>N</sup> <sup>¼</sup> <sup>60</sup>Ns 2π

2 4

C ffiffiffiffiffiffiffiffiffiffiffiffiffi gcRT<sup>0</sup> p ¼

Static density can be calculated from the following correlations

r0 rst

The rotational speed can be calculated using the following correlation

where, d2, the rotor diameter can be calculated using the correlation

limit of the ratio of the outlet to inlet velocity ratio as 0.8.

2.6. Density at the inlet of the blade

2.7. Rotational speed

calculations

2.8. Blade axial width at outlet

2.9. Flow separation condition

For the sake of easy reference, even number of blades are chosen so that half of the blades can be considered as splitter blades. In general for a better design, 20 blades and 45� blade angle at outlet are taken.

### 2.5. Blade peripheral speed

The peripheral speed of the impeller is calculated using the following correlations

$$-\Delta h\_0 = \psi u^2 \tag{4}$$

where increase in the enthalpy is given by

$$
\Delta h\_0 = \mathbb{C}\_p (T\_{0,2} - T\_{0,1}) \tag{5}
$$

The isentropic law can be applied, which is given by

$$\frac{T\_{0,2}}{T\_{0,1}} = \left(\frac{p\_{0,2}}{p\_{0,1}}\right)^{\left[\left(\frac{R}{C\_p}\right)\frac{1}{\eta\_{p,c}}\right]}\tag{6}$$

The polytropic efficiency ηp, <sup>c</sup> can be calculated by choosing isentropic efficiency ηs, <sup>c</sup> as 0.83 and optimum specific speed Ns as 0.8 for the better design and using the following correlation

<sup>η</sup>s, <sup>c</sup> <sup>¼</sup> <sup>r</sup> <sup>R</sup>=Cp �<sup>1</sup> <sup>r</sup> <sup>R</sup>=Cp <sup>Þ</sup>ηp, <sup>c</sup>�<sup>1</sup> � (7)

Other blade parameters can be calculated using the regular trigonometric correlations from the velocity triangles of the blade like the following

Numerical Simulations of a High-Resolution RANS-FVDM Scheme for the Design of a Gas Turbine Centrifugal… http://dx.doi.org/10.5772/intechopen.72098 101

$$\mathbb{C}\_{u,2,ac} = \psi u\_2 \tag{8}$$

$$\mathbb{C}\_{r,2} = \frac{\mathbb{C}\_{u,2,ac}}{\tan \alpha\_{c,2}} \tag{9}$$

$$\mathbf{C\_2} = \frac{\mathbf{C\_{u,2,ac}}}{\sin \alpha\_{c,2}} \tag{10}$$

### 2.6. Density at the inlet of the blade

where number of blades is denoted by Z and β<sup>2</sup> represents the angle between the radial

The number of blades and the exit blade angle are dependent on each other and can be calculated using loading coefficient. Loading coefficient ψ is the ratio between the outlet

> <sup>¼</sup> tanβ<sup>2</sup> tanαc, <sup>2</sup> � �

For the sake of easy reference, even number of blades are chosen so that half of the blades can

þ 1 σw

�Δh<sup>0</sup> <sup>¼</sup> <sup>ψ</sup>u<sup>2</sup> (4)

Δh<sup>0</sup> ¼ Cpð Þ T0,<sup>2</sup> � T0, <sup>1</sup> (5)

� (7)

(3)

(6)

blade angle at

� ��<sup>1</sup>

tangential flow velocity and blade speed, which is given by the correlation

be considered as splitter blades. In general for a better design, 20 blades and 45�

The peripheral speed of the impeller is calculated using the following correlations

T0, <sup>2</sup> T0, <sup>1</sup> <sup>¼</sup> <sup>p</sup>0,<sup>2</sup> p0,<sup>1</sup>

<sup>η</sup>s, <sup>c</sup> <sup>¼</sup> <sup>r</sup>

! <sup>R</sup>

The polytropic efficiency ηp, <sup>c</sup> can be calculated by choosing isentropic efficiency ηs, <sup>c</sup> as 0.83 and optimum specific speed Ns as 0.8 for the better design and using the following correlation

Other blade parameters can be calculated using the regular trigonometric correlations from the

<sup>R</sup>=Cp �<sup>1</sup> <sup>r</sup> <sup>R</sup>=Cp <sup>Þ</sup>ηp, <sup>c</sup>�<sup>1</sup>

Cp � � <sup>1</sup> <sup>η</sup>p, <sup>c</sup> h i

<sup>ψ</sup> <sup>¼</sup> Cu,2,sc u2

such that the downstream diffuser does not get prone to stall when

direction and tangent to the rotor blade at the periphery.

C2and Cr, <sup>2</sup> are reduced by keeping very large αc, 2.

αc,<sup>2</sup> should be less than 60�

100 Numerical Simulations in Engineering and Science

2.4. Number of blades

outlet are taken.

2.5. Blade peripheral speed

where increase in the enthalpy is given by

The isentropic law can be applied, which is given by

velocity triangles of the blade like the following

The Mach number at the inlet can be calculated using the following correlation

$$\frac{\mathcal{C}}{\sqrt{g\_c R T\_0}} = \sqrt{2 \frac{\mathcal{C}\_p}{R} \left[ 1 - \left[ 1 + \frac{M^2}{2\left(\frac{\mathcal{C}\_p}{R} - 1\right)} \right]^{-1} \right]} \tag{11}$$

Static density can be calculated from the following correlations

$$\frac{\rho\_0}{\rho\_{st}} = \left[1 + \frac{M^2}{2\left(\frac{C\_r}{R} - 1\right)}\right]^{\left(\frac{C\_p}{R} - 1\right)}\tag{12}$$

### 2.7. Rotational speed

The rotational speed can be calculated using the following correlation

$$N = \frac{60 \text{N}\_s}{2\pi} \frac{\left(\text{g}\_c \Delta h\_0\right)^{\circ\_4}}{\sqrt{\dot{V}}} \tag{13}$$

### 2.8. Blade axial width at outlet

The outlet blade axial width is calculated with the assumptions that the effect of the thickness and boundary layer of the blade are neglected. The following correlations are used for the calculations

$$
\dot{m} = \mathcal{C}\_{r,2} \rho\_{st,2} \pi d\_2 b\_2 \tag{14}
$$

where, d2, the rotor diameter can be calculated using the correlation

$$
\mu\_2 = \frac{\pi d\_2 N}{60} \tag{15}
$$

### 2.9. Flow separation condition

Flow separation should be minimized in the design. To reduce the separation of the flow, the back sweep angle should be higher. Furthermore, the separation in the rotor with subsonic inlet relative Mach number and normal Reynold's number can be minimized by using a lower limit of the ratio of the outlet to inlet velocity ratio as 0.8.
