Air (assumed incompressible) flows across the rotor shown in Fig. P 12.2 such that the magnitude

step 1 This exercise if sine theta is equal to 1 .9 times 15 times cosine 30 degrees divided by 1 .2 ti

Air (assumed incompressible) flows across the rotor shown in...

Air (assumed incompressible) flows across the rotor shown in Fig. $\mathrm{P} 12.2$ such that the magnitude of the absolute velocity increases from $15 \mathrm{m} / \mathrm{s}$ to $25 \mathrm{m} / \mathrm{s}$. Measurements indicate that the absolute velocity at the inlet is in the direction shown. Determine the direction of the absolute velocity at the outlet if the fluid puts no torque on the rotor. Is the rotation $\mathrm{CW}$ or $\mathrm{CCW}$ ? Is this device a pump or a turbine?

Key Concepts

Incompressible Flow

Assuming incompressible flow simplifies the analysis by keeping the density constant throughout the flow. This assumption is often valid for liquids and low-speed gas flows, and it eliminates the need to account for density variations in the energy and momentum equations.

Euler’s Turbomachinery Equation

Euler’s turbomachinery equation links the work transferred in the rotor to the change in angular momentum of the fluid. It is a foundational relationship in analyzing and designing devices like pumps and turbines. Depending on the sign of the work exchange, the device can be identified as a pump (imparting energy to the fluid) or a turbine (extracting energy from the fluid).

Torque and Device Classification in Turbomachinery

The concept of torque in turbomachinery is tied to the exchange of angular momentum between the fluid and the rotor. When the fluid puts no net torque on the rotor, it indicates that the change in tangential momentum is balanced. This balance, along with the direction of the work transfer deduced from the velocity components, helps in determining whether the device is acting as a pump or as a turbine, as well as providing insight into the rotation direction of the rotor.

Conservation of Angular Momentum

In fluid flow through rotors, the conservation of angular momentum implies that if no net torque is applied to the rotor, then the change in the fluid’s angular momentum between the inlet and outlet must be zero. This principle is used to relate the tangential components of the velocity entering and leaving the rotor and is essential in assessing the rotational effects on the flow.

Velocity Triangles

Velocity triangles are a graphical tool used in turbomachinery to relate the absolute velocity, the relative velocity (relative to the moving blades), and the blade velocity. These triangles help in understanding the interaction between the fluid flow and the rotor blades, particularly in determining the direction and magnitude of velocity components that influence the work transfer.

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