## Conservation of momentum

Newton's second law of motion may be defined for fluids as: 'At any instant in steady flow, the resultant force acting on the moving fluid within a fixed volume of space equals the net rate of outflow of momentum from the closed surface bounding that volume.' This is known as the momentum theorem.

As an example, consider a fluid passing across a turbine in a pipe. In Figure 2.3, fluid flowing at speed u1 into the left of the control surface carries momentum pu1 per unit volume in the direction of flow. In time At, the volume entering the surface is A1u1At. Therefore the rate at which the momentum is entering the control surface along the x direction is

Similarly the rate at which momentum is leaving the control volume is pA1u:2. The momentum theorem tells us that the rate of change of momentum equals the force, F on the fluid and the reaction, — F is the force exerted on the turbine and pipe by the fluid. So

Control

Area A1

Control

Area A1

Figure 2.3 A turbine in a pipe. The dotted line shows the control surface over which the momentum theorem is applied.

Figure 2.3 A turbine in a pipe. The dotted line shows the control surface over which the momentum theorem is applied.

where m = pA1u1 = pA2u2 is the mass flow and the signs indicate directions, which are obvious in this case. However, in more complex cases, such as inside a turbine, it is important to remember that momentum and force are vectors (i.e. their direction matters!).

## Renewable Energy 101

Renewable energy is energy that is generated from sunlight, rain, tides, geothermal heat and wind. These sources are naturally and constantly replenished, which is why they are deemed as renewable. The usage of renewable energy sources is very important when considering the sustainability of the existing energy usage of the world. While there is currently an abundance of non-renewable energy sources, such as nuclear fuels, these energy sources are depleting. In addition to being a non-renewable supply, the non-renewable energy sources release emissions into the air, which has an adverse effect on the environment.

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