## M

where I is the moment of inertia of the body with respect to the center of rotation.

For a fixed mass, the conservation of linear momentum is equivalent to Newton's second law:

The corresponding expression for the conservation of angular momentum is

where is the moment (torque) acting on the system and da/dt = « is the angular acceleration.

For a flow system, streams with curved streamlines may carry angular momentum into and/or out of the system by convection. To account for this, the general macroscopic angular momentum balance applies:

in out

For a steady-state system with only one inlet and one outlet stream, this becomes

This is known as the Euler turbine equation, because it applies directly to turbines and all rotating fluid machinery. We will find it useful later in the analysis of the performance of centrifugal pumps.

### D. Moving Boundary Systems and Relative Motion

We sometimes encounter a system that is in contact with a moving boundary, such that the fluid that composes the system is carried along with the boundary while streams carrying momentum and/or energy may flow into and/or out of the system. Examples of this include the flow impinging on a turbine blade (with the system being the fluid in contact with the moving blade) and the flow of exhaust gases from a moving rocket motor. In such cases, we often have direct information concerning the velocity of the fluid relative to the moving boundary (i.e., relative to the system), Vr, and so we must also consider the velocity of the system, Vs, to determine the absolute velocity of the fluid that is required for the conservation equations.

For example, consider a system that is moving in the x direction with a velocity Vs a fluid stream entering the system with a velocity in the x direction relative to the system of Vri, and a stream leaving the system with a velocity Vro relative to the system. The absolute stream velocity in the x direction Vx is related to the relative velocity Vrx and the system velocity Vsx by

The linear momentum balance equation becomes

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