Bernoulli and energy equations

The Bernoulli equation is a relation between pressure, velocity, and elevation in steady, incompressible flow, and is expressed along a streamline and in regions where net viscous force are negligible as

$\frac{P}{\rho }+\frac{v^2}{2}+gz=cons\tan t$

The Bernoulli equation can also be considered as an expression of mechanical energy balance, stated as the sum of the kinetic, potential, and flow energies of a fluid particle is constant along a streamline during steady flow when the compressibility and frictional effects are negligible. Multiplying the Bernoulli equation by density gives:

$P+\rho \frac{V^2}{2}+\rho gz=cons\tan t$

where P is the static pressure, which represents the actual pressure of the fluid; $\rho \frac{V^2}{2}$is the dynamic pressure, which represents the pressure rises when the fluid in motion is brought to a stop; and $\rho gz$ is the hydrostatic pressure, whichh accounts for the effects of fluid weight on pressure. The sum of the static, dynamic, and hydrostatic pressures is called the total pressure. The Bernoulli equation states that the total pressure along a streamline is constant. The sum of the static and the dynamic pressures is called the stagnation pressure, which represents complete stop in a frictionless manner. The Bernoulli equation can also be represented in terms of "heads" by dividing each term by g.

$\frac{P}{\rho g}+\frac{V^2}{2g}+z=H=cons\tan t$

where $\frac{P}{\rho g}$ is the pressure head, which represents the height of a fluid column that produces the static pressure P; $\frac{V^2}{2g}$ is the velocity head, which represents the elevation needed for a fluid to reach the velocity V during frictionless free fall; and z is the elevation head, which represents the potential energy of the fluid. Also, H is the total head for the flow. The curve that represents the sum of the static pressure and the elevation heads, 

$\frac{P}{\rho g}+z,$ is called the hydraulic grade line (HGL), and the curve that represents the total head of the fluid, $\frac{P}{\rho g}+\frac{V^2}{2g}+z$，is called the energy grade line (EGL).

The energy equation for steady, incompressible flow is expressed as

$$\begin{gathered} \frac{P_1}{\rho g}+{\alpha }_1\frac{V_1^2}{2g}+z_1+h_{pump,u}=\frac{P_2}{\rho g}+{\alpha }_2\frac{V_2^2}{2g}+z_2+h_{turbine,e}+h_L \\ where \\ {h}_{pump,u}=\frac{w_{pump,u}}{g}=\frac{{\dot{W}}_{pump,u}}{\dot{m}g}=\frac{{\eta }_{pump}{\dot{W}}_{pump}}{\dot{m}g} \\ {h}_{turbine,e}=\frac{w_{turbine,e}}{g}=\frac{{\dot{W}}_{turbine, e}}{\dot{m}g}=\frac{{\dot{W}}_{turbine}}{{\eta }_{turbine}\dot{m}g} \\ {e}_{mech,loss}=u_2-u_1-q_{net,in} \end{gathered}$$

The Bernoulli and energy equation are two of the most fundamental relations in fluid mechanics.