# Axial Flow Compressors - Mean Line Analysis

Preliminary design of axial compressors generally consist of a mean-line analysis, radial variations of airflow are ignored. The analysis is conducted at the mean blade radius location.

Figure 1: Mean line of flow across the compressor stage.

The ideal flow of fluid through a compressor stage can be illustrated by velocity triangle diagrams as shown in Figure 2. The fluid flow is assumed to travel along a path that follows the blade curvatures.

Figure 2: Compressor stage velocity diagram.

The fluid can be assumed incompressible if the Mach number is less than 0.3, which is a good first assumption to make for small scale EDFs. For incompressible fluids, density is considered constant. This means that, like mass flow rate, the volume flow rate at any location along a duct is constant. Thus, assuming constant duct cross-sectional area across the stages and infinitely thin blades, the axial velocity across the compressor stage $C_x$ will be constant.

In Figure 2, the rotor blade at the mean radius travels at a speed $U$, which is equal to its angular velocity $\omega$ multiplied by the mean radius of the blade $r$. The velocity $U$ is radial and perpendicular to the axial direction. The absolute velocity of the fluid at the entrance of the rotor, exit of the rotor (entrance of the stator) and exit of the stator are $C_1$, $C_2$ and $C_3$ respectively. $W_1$ and $W_2$ are the fluid velocities relative to the moving rotor blade. The relative fluid velocity follows the contours of the blades.

#### Performance predictions

##### Mass Flow Rate

$$\rho C_x A_f$$

##### Thrust

$$\mathbf{F}_{t}=\dot{m}\mathbf{v}$$

##### Power

From Bernoulli's equation for adiabatic flow, the power, $P$, supplied to the fluid by the compressor stage is the change in static enthalpy across the rotor:

$$P=h_{02}-h_{01}$$

where $h_{01}$ and $h_{02}$ are the static enthalpies at the rotor entrance and exit respectively.

The power is also equal to the torque acting on the rotor multiplied by its angular velocity:

$$P=\tau\omega$$

where $\tau$ is the torque acting on the rotor and $\omega$.

The torque acting on the rotor is equal to the rate of change of the momentum of the flow through the rotor:

$$\tau=\frac{d}{dt} (mC_\theta)=\dot{m} (C_{\theta 1} - C_{\theta 2})r_{m}$$

thus the power require to propel fluid axially through the fan at a speed of $Cx$ becomes:

$$P=\dot{m}U(C_{\theta 1} - C_{\theta 2})$$

where

$$\omega = \frac{U}{r_m}$$

#### Preliminary compressor stage design

There are three non-dimensional numbers that describes the performance of a compressor stage: the Stage Loading $\psi$, the Flow Coefficient $\phi$ and the Reaction $R$.

The flow coefficient is ratio of the axial flow and the blade's tangential velocity:

$\phi=\frac{C_x}{U}$ (1)

The stage loading coefficient describes the change in stagnation enthalpy divided by the square of the rotor speed:

$\psi=\frac{h_{03}-h_{01}}{U^2}=\frac{C_{\theta 1} - C_{\theta 2}}{U}=\phi(tan(\beta_1)-tan(\beta_2))$ (2)

The compressor's Reaction is the ratio of the rotor static enthalpy rise to the stage static enthalpy rise. It expresses the how much of the enthalpy rise occurs through the rotor compared to the stator.

$R=\frac{h_2-h_1}{h_3-h_1} = \frac{1}{2}\phi(\tan(\beta_1) + \tan(\beta_2))$ (3)

Solving equation (2) for $\tan(\beta_1)$:

$\frac{\psi}{\phi} + \tan(\beta_2) = \tan(\beta_1)$ (4)

substituting (4) into (3):

$\frac{2R}{\phi} = \frac{\psi}{\phi} + 2\tan{\beta_2}$ (5)

Solving (5) for $\beta_2$:

$\beta_2 = \tan^{-1}(\frac{R-\frac{\psi}{2}}{\phi})$ (6)

and then substituting (6) into (4) and solving for $$\beta_1: \beta_1 = \tan^{-1}(\frac{\psi}{\phi} + \frac{2R - \psi}{2\phi}) (7) This sets the geometry of the rotor blades. The geometry of the stator blades is defined from the velocity triangles as:$$\alpha_1=\tan^{-1}(\frac{C_{\theta 1}}{C_x})\alpha_2=\tan^{-1}(\frac{C_{\theta 2}}{C_x})$$where$$C_{\theta 1} = U-W_1\sin(\beta_1)C_{\theta 2} = U-W_2\sin(\beta_2)$$and$$W_1 = \frac{C_x}{cos(\beta_1)}W_2 = \frac{C_x}{cos(\beta_2)}

-pitch to chord ratios between 0.8 and 1.2 [Describe how to choose pitch chord ratio pg 154 eq5.18]
-Flow coefficient between 0.4 and 0.8. High flow coefficients increase Ma number which can cause problems. Higher flow coefficient stages are less tolerance of inflow disturbances.

#### Low reaction vs. high reaction compressor stage

Reaction is one of three non-dimensional numbers used to describe a compressor stage. It is defined as the enthalpy change over the rotor divided by the enthalpy change over the entire stage. Basically it expresses where the work done in moving the fluid is done: across the rotor or across the stator.

Shown in the Figures 1 and 2 are example blade geometry for a low reaction compressor stage where the work of moving the flow occurs on the stator blades and a high reaction compressor stage where the work done of moving the flow occurs on the rotor blades. Note that high reaction compressor blades have the exit flow that is mostly axial. This type of compressor stage is ideal for a single stage ducted fan. The lower the reaction the higher the whirl velocity at the inlet and outlet of the stage.

#### References:

[1] "Fluid Mechanics and Thermodynamics of Turbomachinery", S. L. Dixon and C. A. Hall, 2010, Hall, 6th ed.