**Bernoulli’s equation**can be considered 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** **effect** causes the **lowering of fluid pressure (static pressure – p)** in regions where the flow velocity increases. The Bernoulli equation can be modified to take into account **gains and losses of the head**. The resulting equation referred to as the **extended Bernoulli’s equation** is useful in solving most fluid flow problems.

**Bernoulli’s equation** has some restrictions in its applicability, they summarized in the 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**. **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 the 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.

## Derivation of Bernoulli’s Equation

### Conservation of Energy

This principle is generally known as the **conservation of energy principle** and states that the **total energy** of an isolated system remains constant — it is said to be conserved over time. This is equivalent to the **First Law of Thermodynamics**, which develops the general energy equation in thermodynamics. This principle can be used in the analysis of **flowing fluids,** and this principle is expressed mathematically by the following equation:

where h is enthalpy, k is the thermal conductivity of the fluid, T is temperature, and Φ is the viscous dissipation function.

### Derivation

**Bernoulli’s equation** for incompressible fluids can be derived from **Euler’s equations** of motion under rather severe restrictions.

- The velocity must be derivable from a
**velocity potential**. - External forces must be conservative. That is, derivable from a potential.
- The density must either be constant or a function of the pressure alone.
- Thermal effects, such as natural convection, are ignored.

The Euler equations are a set of quasilinear hyperbolic equations governing adiabatic and inviscid flow in fluid dynamics. Euler equations can be obtained by linearization of these Navier–Stokes equations.

## Extended Bernoulli’s Equation

Two main **assumptions** were applied to the derivation of the **simplified Bernoulli equation**.

- The first restriction on Bernoulli’s equation is that
**no work is allowed**to be done on or by the fluid. This is a significant limitation because most hydraulic systems (especially in nuclear engineering) include pumps. This restriction prevents two points in a fluid stream from being analyzed if a pump exists between the two points.

- The second restriction on simplified Bernoulli’s equation is that
**no fluid friction**can solve hydraulic problems. In reality,**friction plays a crucial role**. The total head possessed by the fluid cannot be transferred completely and is lossless from one point to another. In reality, one purpose of pumps incorporated in a hydraulic system is to overcome the losses in pressure due to friction.

Due to these restrictions, most of the practical applications of the simplified **Bernoulli equation** to real hydraulic systems are very limited. The simplified **Bernoulli equation must be modified** to deal with both head losses and pump work.

The Bernoulli equation can be modified to take into account **gains and losses of the head**. The resulting equation referred to as the **extended Bernoulli’s equation** is useful in solving most fluid flow problems. The following equation is one form of the extended Bernoulli 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 pipe walls. 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 not represent a loss of total energy** of the fluid. The total energy of the fluid is conserved 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 fluid’s internal energy** (temperature increases).

Most methods for evaluating head loss due to friction are based almost exclusively on experimental evidence. This will be discussed in the following sections.

## Example – Relation between Pressure and Velocity

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

When **Bernoulli’s equation** is combined with the continuity equation, the two can 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 with 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 the bottom of a **reactor core**, about 5 meters below the cold leg of primary piping.

Let assume:

- The fluid of constant density
**⍴ ~ 720 kg/m**(at 290°C) flows steadily through the cold leg and 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 **Bernoulli’s** principle, the pressure at the bottom of the core (core inlet) is:

## Bernoulli’s Principle – Lift Force

In general, **the lift** is an upward-acting force on an aircraft wing or **airfoil**. There are several ways to explain **how an airfoil generates lift**. Some theories are more complicated or more mathematically rigorous than others. Some theories are incorrect. There are theories based on **Bernoulli’s principle,** and theories are based directly on **Newton’s third law**.

The explanation based on **Newton’s third law** states that the lift is caused by a **flow deflection** of the airstream behind the airfoil. The airfoil generates lift by exerting a downward force on the air as it flows past. According to Newton’s third law, the air must **exert an upward force on the airfoil**. This is a very simple explanation.

**Bernoulli’s principle** combined with the **continuity equation** can also be used to determine the lift force on an airfoil if the behavior of the fluid flow in the vicinity of the foil is known. In this explanation, the **shape** of an airfoil is crucial. The shape of an airfoil causes air to **flow faster on top** than on the bottom. According to **Bernoulli’s principle**, faster-moving air exerts **less pressure**, and therefore the air must exert an **upward force** on the airfoil (as a result of a pressure difference).

**Bernoulli’s principle** requires airfoil to be of an **asymmetrical shape**. Its surface area must be **greater on the top** than on the bottom. As the air flows over the airfoil, it is displaced more by the top surface than the bottom. According to the continuity principle, this displacement must **increase flow velocity **(resulting in a decrease in pressure). The flow velocity is increased by the bottom airfoil surface but considerably less than the flow on the top surface. The lift force of an airfoil, characterized by the** lift coefficient**, can be changed during the flight by changes in the shape of an airfoil. The lift coefficient can thus be even doubled with relatively simple devices (**flaps and slats**) if used on the full span of the wing.

The use of **Bernoulli’s principle** may not be correct. Bernoulli’s principle assumes **incompressibility** of the air, but in reality, the air is easily compressible. But there are more limitations of explanations based on Bernoulli’s principle. There are two main popular explanations of lift:

- Explanation based on downward deflection of the flow –
**Newton’s third law** - Explanation based on changes in flow speed and pressure –
**Continuity principle and Bernoulli’s principle**

Both explanations correctly identify some aspects of the lift forces but leave other important aspects of the phenomenon unexplained. A more comprehensive explanation involves both changes in flow speed and downward deflection and requires looking at the flow in more detail.

See more: Doug McLean, *Understanding Aerodynamics: Arguing from the Real Physics.* John Wiley & Sons Ltd. 2013. ISBN: 978-1119967514

## Bernoulli’s Effect – Spinning ball in an airflow

**Bernoulli’s effect** has another interesting consequence. Suppose a** ball** is **spinning** as it travels through the air. As the ball spins, the surface friction of the ball with the surrounding air drags a thin layer (referred to as the** boundary layer**) of air with it. It can be seen from the picture the boundary layer is on one side traveling in the **same direction** as the air stream that is flowing around the ball (the upper arrow). On the other side, the boundary layer travels in the **opposite direction** (the bottom arrow). On the side of the ball where the air stream and boundary layer are moving in the opposite direction (the bottom arrow) to each other, friction between the two **slows the air stream**. On the opposite side, these layers move in the same direction, and the **stream moves faster**.

According to **Bernoulli’s principle**, faster-moving air exerts less pressure, and therefore the air must exert an upward force on the ball. In fact, in this case, the use of Bernoulli’s principle may not be correct. Bernoulli’s principle assumes incompressibility of the air, but in reality, the air is easily compressible. But there are more limitations of explanations based on Bernoulli’s principle.

In Robert G. Watts and Ricardo Ferrer (The lateral forces on a spinning sphere: Aerodynamics of a curveball), this effect can be explained by another model, which gives important attention to the spinning boundary layer of air around the ball. The boundary layer tends to separate prematurely on the side of the ball, where the air stream and **boundary layer** are moving in the opposite direction (the bottom arrow). On the side of the ball, where the air stream and boundary layer are moving in the same direction, the boundary layer carries further around the ball before it separates into turbulent flow. This gives a **flow deflection** of the airstream in one direction behind the ball. The rotating ball generates lift by exerting a downward force on the air as it flows past. According to **Newton’s third law**, the air must exert an upward force on the ball.

## Torricelli’s law

**Torricelli’s law**, also known as **Torricelli’s principle**, or **Torricelli’s theorem**, statement in fluid dynamics that the speed, v, of fluid flowing out of an** orifice** under the force of gravity in a tank is proportional to the square root of the vertical distance, h, between the liquid surface and the center of the orifice and to the square root of twice the acceleration caused by gravity (g = 9.81 N/kg near the surface of the earth).

In other words, the efflux velocity of the fluid from the orifice is the same as that it would have acquired by falling a height h under gravity. The law was discovered by and named after the Italian scientist **Evangelista Torricelli** in 1643. It was later shown to be a particular case of **Bernoulli’s principle**.

**Torricelli’s equation** is derived for a specific condition. The orifice must be small, and viscosity and other losses must be ignored. If a fluid is flowing through a very small orifice (for example, at the bottom of a large tank), then the velocity of the fluid at the large end can be neglected in Bernoulli’s Equation. Moreover, the speed of efflux is independent of the direction of flow. In that case, the efflux speed of fluid flowing through the orifice is given by the following formula:

v = √2gh