## Bernoulli’s Equation

**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**. **Bernoulli’s equation** has some restrictions in its applicability, they summarized in following points:

- steady flow system,
- density is constant (which also means the fluid is incompressible),
- no work is done on or by the fluid,
- no heat is transferred to or from the fluid,
- no change occurs in the internal energy,
- the equation relates the states at two points along a single streamline (not conditions on two different streamlines)

Under these conditions, 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.

## Bernoulli’s Effect – Relation between Pressure and Velocity

It is an illustrative example, following data **do not** correspond to any reactor design.

When the **Bernoulli’s equation** is combined with the continuity equation the two can be used to find velocities and pressures at points in the flow connected by a streamline.

**The continuity equation** is simply a mathematical expression of the principle of conservation of mass. For a control volume that has a **single inlet** and a **single outlet**, the principle of conservation of mass states that, for **steady-state flow**, the mass flow rate into the volume must equal the mass flow rate out.

**Example:**

**Determine pressure and velocity** within a cold leg of primary piping and determine pressure and velocity at a bottom of a **reactor core**, which is about 5 meters below the cold leg of primary piping.

Let assume:

- Fluid of constant density
**⍴ ~ 720 kg/m**(at 290°C) is flowing steadily through the cold leg and through the core bottom.^{3}

- Primary piping flow cross-section (single loop) is equal to
**0.385 m**(piping diameter ~ 700mm)^{2}

- Flow velocity in the cold leg is equal to
**17 m/s**.

- Reactor core flow cross-section is equal to
**5m**.^{2}

- The gauge pressure inside the cold leg is equal to
**16 MPa**.

As a result of the Continuity principle the velocity at the bottom of the core is:

v_{inlet} = v_{cold} . A_{piping} / A_{core} = 17 x 1.52 / 5 = **5.17 m/s**

As a result of the **Bernoulli’s** principle the pressure at the bottom of the core (core inlet) is:

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