## Conservation of Energy in Fluid Mechanics – Bernoulli’s Principle

The law of conservation of energy can be used also in the analysis of **flowing ****fluids**.

**The Bernoulli’s equation** can be considered to be a statement of the **conservation of energy principle** appropriate for flowing fluids. It is one of the most important/useful equations in **fluid mechanics**. It puts into a relation **pressure and velocity** in an **inviscid incompressible flow**. The general energy equation is simplified to:

This equation is the most famous equation in **fluid dynamics**. **The Bernoulli’s equation** describes the qualitative behavior flowing fluid that is usually labeled with the term **Bernoulli’s effect**. This effect causes the **lowering of fluid pressure** in regions where the flow velocity is increased. This lowering of pressure in a constriction of a flow path may seem counterintuitive, but seems less so when you consider pressure to be** energy density**. In the high velocity flow through the constriction, **kinetic energy** must increase at the expense of **pressure energy**. The dimensions of terms in the equation are kinetic energy per unit volume. The equation above assumes that no **non-conservative forces** (e.g. friction forces) are acting on the fluid. This is very strong assumption.

**Extended Bernoulli’s equation**

The Bernoulli’s equation can be modified to take into account **gains and losses of head, **caused by **external forces** and **non-conservative forces**. The resulting equation, referred to as the **extended Bernoulli’s equation**, is very useful in solving most fluid flow problems. The following equation is one form of the extended Bernoulli’s equation.

where:

- h = height above reference level (m)
- v = average velocity of fluid (m/s)
- p = pressure of fluid (Pa)
- H
_{pump}= head added by pump (m) - H
_{friction}= head loss due to fluid friction (m) - g = acceleration due to gravity (m/s
^{2})

**The head loss** (or the pressure loss) due to fluid friction (H_{friction}) represents the energy used in overcoming friction caused by the walls of the pipe. The head loss that occurs in pipes is dependent on the **flow velocity, pipe diameter **and** length**, and a **friction factor** based on the roughness of the pipe and the **Reynolds number** of the flow. A piping system containing many pipe fittings and joints, tube convergence, divergence, turns, surface roughness and other physical properties will also increase the head loss of a hydraulic system.

Although the **head loss represents a loss of energy**, it does **does not represent a loss of total energy** of the fluid. The total energy of the fluid conserves as a consequence of the **law of conservation of energy**. In reality, the head loss due to friction results in an equivalent **increase in the internal energy** (increase in temperature) of the fluid.

This phenomenon can be seen also in case of reactor coolant pumps. Generally reactor coolant pumps are very powerful, they can consume **up to 6 MW each** and therefore they can be used for heating the primary coolant before a reactor startup. For example from 30°C at cold zero power (CZP) up to 290°C at hot zero power (HZP).

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