Continuity equation basis and Bernoulli equation

Basis of the continuity equation

The continuity equation is derived from the law of conservation of mass: for steady flow, mass entering a control volume per second equals mass leaving it. For incompressible flow, $Q = A_1V_1 = A_2V_2$ (in general $\rho_1A_1V_1 = \rho_2A_2V_2$).

Bernoulli's equation

It is based on the conservation of energy (derived from Euler's equation of motion). For steady, ideal, incompressible flow along a streamline, the total energy per unit weight is constant:

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

Between two sections 1 and 2:

$$\frac{p_1}{\rho g} + \frac{V_1^2}{2g} + z_1 = \frac{p_2}{\rho g} + \frac{V_2^2}{2g} + z_2$$

• $p/\rho g$ = pressure head (pressure energy per unit weight)
• $V^2/2g$ = velocity head (kinetic energy per unit weight)
• $z$ = datum head (potential energy per unit weight)

  --------------------------------- EGL
   V1^2/2g             V2^2/2g (larger)
  ---------._____
                 '-.______________ HGL
   p1/rho g              p2/rho g
      |    ____________      |
  ====|===/    pipe    \_____|=====
  (1) | z1             (2)   | z2
  --------------- datum -------------

At the narrow section 2, $V_2 > V_1$ (continuity), so the velocity head rises and the pressure head falls, while the total stays the same.

For real fluids: $\dfrac{p_1}{\rho g} + \dfrac{V_1^2}{2g} + z_1 = \dfrac{p_2}{\rho g} + \dfrac{V_2^2}{2g} + z_2 + h_L$.

Assumptions: ideal, steady, incompressible, irrotational flow along a streamline; only gravity and pressure forces act.

Applications: venturimeter, orifice meter, pitot tube, flow through nozzles, net head of a hydropower plant.