---
title: "Continuity equation basis and Bernoulli equation"
subject: "Hydro Power"
chapter: 1
key: "semester_6/hydro_power:continuity-and-bernoulli"
source: https://www.bctengineer.com/q/semester_6/hydro_power/continuity-and-bernoulli
site: BCT Engineering
---

# Continuity equation basis and Bernoulli equation

Hydro Power · Chapter 1: Introduction to Basic Fluid Mechanics and Hydraulics

- Asked in: 2080 Chaitra, 1+5 marks
- Video: [Understanding Bernoulli's Equation](https://www.youtube.com/watch?v=DW4rItB20h4) — The Efficient Engineer
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_6/hydro_power/continuity-and-bernoulli
- Whole chapter: https://www.bctengineer.com/md/semester_6/hydro_power/chapter-1

## Question

In which basis the continuity equation is derived? Discuss Bernoulli's equation with mathematical expression and neat sketch.

## Short answer

Continuity equation derives from mass conservation: ρ1A1V1 = ρ2A2V2. Bernoulli's equation from energy conservation: p/ρg + V²/2g + z = constant along streamline. For real fluids add head loss hL. Sketch shows EGL constant, HGL and velocity head varying inversely in converging/diverging pipe.

## Answer

#### 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)

```text
  --------------------------------- 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.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. On which law is the continuity equation based?
   A. Newton's second law
   B. Conservation of mass
   C. Conservation of momentum
   D. Conservation of energy
   Answer: B. Mass entering a control volume equals mass leaving in steady flow: Q = A1V1 = A2V2 for incompressible flow. (Key point: Basis of continuity)

2. Bernoulli's equation is based on which principle?
   A. Second law of thermodynamics
   B. Archimedes' principle
   C. Conservation of mass
   D. Conservation of energy (from Euler's equation)
   Answer: D. It states that the total energy per unit weight is constant along a streamline. (Key point: Basis of Bernoulli)

3. At a narrow section the velocity increases. What happens to pressure head, in ideal flow?
   A. It increases
   B. It stays the same
   C. It becomes zero
   D. It decreases
   Answer: D. Total head is constant; continuity raises V and so the velocity head, hence pressure head falls. (Key point: Narrowing section)

4. What does the term V^2/2g represent?
   A. Potential energy per unit weight
   B. Friction loss
   C. Pressure energy per unit weight
   D. Kinetic energy per unit weight (velocity head)
   Answer: D. p/rho g is pressure head, V^2/2g velocity head, and z datum head. (Key point: Terms)

## Related in Hydro Power

- [Deriving Bernoulli's equation from Euler's equation](https://www.bctengineer.com/question.md?key=semester_6%2Fhydro_power%3Aeuler-bernoulli-derivation) — Hydro Power, chapter 1
- [Bernoulli's equation limitations and application](https://www.bctengineer.com/question.md?key=semester_6%2Fhydro_power%3Abernoulli-limitations-application) — Hydro Power, chapter 1
- [Bernoulli's equation in real practice](https://www.bctengineer.com/question.md?key=semester_6%2Fhydro_power%3Abernoulli-real-practice) — Hydro Power, chapter 1
- [Types of heads and Bernoulli's assumptions](https://www.bctengineer.com/question.md?key=semester_6%2Fhydro_power%3Aheads-bernoulli-assumptions) — Hydro Power, chapter 1, asked 2×
- [Bernoulli for real flow with HGL and EGL](https://www.bctengineer.com/question.md?key=semester_6%2Fhydro_power%3Abernoulli-real-flow-hgl-egl) — Hydro Power, chapter 1

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