## Extended Bernoulli’s Equation

There are two main** assumptions**, that were applied on the derivation of the **simplified Bernoulli’s 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**is allowed in solving hydraulic problems. In reality,**friction plays crucial role**. The total head possessed by the fluid cannot be transferred completely and 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 practical applications of the simplified **Bernoulli’s equation** to real hydraulic systems are very limited. In order to deal with both head losses and pump work, the simplified **Bernoulli’s equation must be modified**.

The Bernoulli equation can be modified to take into account **gains and losses of head**. 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.

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

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