Bernoulli's equation limitations and application

Bernoulli's equation states that, for steady, ideal, incompressible flow along a streamline, the sum of pressure head, velocity head and datum head is constant.

Derivation (from Euler's equation)

Consider a small element of length $ds$ and area $dA$ along a streamline. Forces in the flow direction:

• net pressure force $= p\,dA - (p + dp)\,dA = -dp\,dA$;
• weight component $= -\rho g\,dA\,ds\,\dfrac{dz}{ds} = -\rho g\, dA\, dz$.

Mass × acceleration $= \rho\,dA\,ds \cdot V\dfrac{dV}{ds} = \rho\, dA\, V\,dV$ (steady flow).

           (p+dp)dA
             <--
        [ element ds ]  <- streamline
   p dA -->     |
                v  rho g dA ds

Newton's second law:

$$-dp\,dA - \rho g\,dA\,dz = \rho\,dA\,V\,dV$$ $$\frac{dp}{\rho} + g\,dz + V\,dV = 0 \quad \text{(Euler's equation)}$$

Integrating with $\rho$ constant and dividing by $g$:

$$\frac{p}{\rho g} + \frac{V^2}{2g} + z = \text{constant}$$

Limitations

1. It is valid only for ideal (non-viscous) fluid. Real flows have friction losses, so a loss term $h_L$ must be added.
2. It assumes steady flow; it does not apply directly to unsteady flow such as water hammer or surge.
3. It assumes incompressible flow; not valid for high-speed gas flow.
4. It applies along a single streamline (or to irrotational flow); it assumes uniform velocity across the section, so for real pipes a kinetic energy correction factor $\alpha$ is needed.
5. It does not include energy added or removed by pumps or turbines, or heat transfer, unless those terms are added.
6. It ignores forces other than gravity and pressure, such as surface tension and centrifugal forces in bends.

Application in hydropower

Net head and power of a plant: applying Bernoulli's equation (with losses) between the reservoir surface (1) and the turbine inlet (2):

$$z_1 = \frac{p_2}{\rho g} + \frac{V_2^2}{2g} + z_2 + h_L$$

The head available at the turbine is $H_{net} = H_{gross} - h_L$, and power $P = \eta\rho g Q H_{net}$. For a Pelton wheel, the jet velocity $V = C_v\sqrt{2gH}$ also comes from Bernoulli's equation. It is also used to design the draft tube (pressure at its inlet) and to size intakes and flow-measuring devices (venturimeter) in the plant.