Bernoulli's Equation

Bernoulli's Equation & Continuity — Detailed formulas, worked example and calculator

1) Fundamental formulas

Bernoulli (along a streamline, inviscid, steady):

$$\frac{P_1}{\rho g} \;+\; \frac{v_1^2}{2g} \;+\; h_1 \;=\; \frac{P_2}{\rho g} \;+\; \frac{v_2^2}{2g} \;+\; h_2$$

Continuity (incompressible):

$$Q = v_1 A_1 = v_2 A_2 \quad\text{with}\quad A=\frac{\pi d^2}{4}$$ $$\Rightarrow v_1 d_1^2 = v_2 d_2^2 \quad(\text{if circular pipe and same fluid})$$

Derived useful rearrangement (solve for \(P_2\))

From Bernoulli: $$\frac{P_1}{\rho g}+\frac{v_1^2}{2g}+h_1=\frac{P_2}{\rho g}+\frac{v_2^2}{2g}+h_2$$ Multiply by \(\rho g\) and rearrange: $$\boxed{P_2 = P_1 + \tfrac{1}{2}\rho\,(v_1^2 - v_2^2) + \rho g\,(h_1 - h_2)}$$ (This is the form used to compute pressure at point 2 given known \(P_1, v_1, v_2, h_1, h_2\).)

2) Inputs (change and press Compute)

Units: diameters in m, pressures in Pa (use 1e5 for 100 kPa), velocities in m/s, heights in m, density kg/m³, g m/s².

d₁ (diameter at section 1) d₂ (diameter at section 2) v₁ (velocity at section 1) P₁ (pressure at section 1) — in Pa h₁ (elevation of section 1) h₂ (elevation of section 2) Fluid density ρ (kg/m³) g (m/s²)

Quick results

Detailed step-by-step solution

Notes

• Bernoulli here neglects head loss (no friction) and assumes steady, incompressible flow along a streamline.
• If pipes have roughness/viscous losses, include head loss term \(h_L\) on RHS: add \(+h_L\) where appropriate.
• Pressures used above are absolute or gauge consistently. If P₁ is gauge and P₂ returned is absolute minus same atmospheric value—treat consistently.

Worked numerical example (with the values shown initially)

Given:

• \(d_1 = 0.100\ \mathrm{m},\ d_2=0.050\ \mathrm{m}\)
• \(v_1 = 2.00\ \mathrm{m/s}\)
• \(P_1 = 200{,}000\ \mathrm{Pa}\) (200 kPa)
• \(h_1 = 0.0\ \mathrm{m},\ h_2 = 1.5\ \mathrm{m}\)
• \(\rho = 1000\ \mathrm{kg/m^3},\ g = 9.81\ \mathrm{m/s^2}\)

How to interpret results: The calculator computes cross-sectional areas, flow rate, velocity at section 2 via continuity, velocity/pressure heads, and uses Bernoulli (no losses) to compute \(P_2\). Use consistent units; if you prefer to input pressures in kPa, multiply by 1000 for Pa.

Bernoulli’s Equation and Continuity Equation

A Practical Perspective for Civil Engineers

In civil engineering practice, particularly in water resources, public health engineering, and hydraulic design, two fundamental principles of fluid mechanics are used almost daily — the Continuity Equation and Bernoulli’s Equation. When applied correctly, these two equations allow us to understand and quantify how water (or any incompressible fluid) behaves when it flows through pipes, channels, nozzles, or transitions.

This article explains these equations in a practical manner, illustrates their combined use through a worked example, and highlights their real-world applications in civil engineering projects.

1. Continuity Equation – Conservation of Mass

The Continuity Equation is based on the principle of conservation of mass. For steady flow of an incompressible fluid (such as water), the discharge remains constant along the pipe or conduit:

$$Q = A_1 v_1 = A_2 v_2$$

Where:

• \( Q \) = Discharge (m³/s)
• \( A \) = Cross-sectional area (m²)
• \( v \) = Mean velocity (m/s)

For circular pipes, the area is \( A = \dfrac{\pi d^2}{4} \). Therefore,

$$v_1 d_1^2 = v_2 d_2^2$$

Practical Implication: When the diameter of a pipe reduces, the velocity must increase to maintain the same discharge. This simple relationship is extensively used while designing water supply networks, rising mains, and pump delivery lines.

2. Bernoulli’s Equation – Conservation of Energy

Bernoulli’s Equation is essentially a statement of conservation of energy for steady, incompressible, and inviscid flow along a streamline. It states that the total energy head remains constant:

$$\frac{P_1}{\rho g} + \frac{v_1^2}{2g} + h_1 = \frac{P_2}{\rho g} + \frac{v_2^2}{2g} + h_2$$

Where:

• \( \dfrac{P}{\rho g} \) = Pressure head (m)
• \( \dfrac{v^2}{2g} \) = Velocity head (m)
• \( h \) = Elevation head (m)

In most practical civil engineering problems, we rearrange Bernoulli’s equation to find the unknown pressure at a downstream section:

$$P_2 = P_1 + \frac{1}{2}\rho (v_1^2 - v_2^2) + \rho g (h_1 - h_2)$$

This form is extremely useful when we know the upstream conditions and geometry, and need to estimate pressure at another location.

3. Worked Example – Interpretation

Consider a pipeline in which water flows from a 100 mm diameter section to a 50 mm diameter section. The velocity at the larger section is 2.0 m/s, pressure is 200 kPa, and there is a rise of 1.5 m in elevation.

Using Continuity:

• Area reduces to one-fourth
• Velocity at the smaller section becomes 8.0 m/s

Using Bernoulli:

• The increase in velocity head causes a significant drop in pressure head
• The elevation rise further reduces the pressure
• The resulting pressure at the smaller section is considerably lower than the upstream pressure

This example clearly demonstrates a common phenomenon observed in pipelines — pressure reduces when velocity increases (and vice versa), provided elevation changes are also accounted for.

4. Practical Applications in Civil Engineering

Application Area	How Bernoulli + Continuity are Used
Water Supply Networks	Sizing of pipes, calculation of residual pressure at consumer end
Pumping Mains	Determination of delivery pressure and selection of pump head
Venturimeter / Orificemeter	Measurement of discharge in pipelines
Spillways & Outlets	Estimation of velocity and pressure over crest or through sluices
Pipe Transitions & Reducers	Checking for negative pressure (cavitation risk)
Fire Fighting Systems	Ensuring adequate residual pressure at hydrants
Irrigation Canals & Pipelines	Design of transitions and drop structures

In rising mains, engineers frequently use these equations to check whether the pressure at high points remains positive. Negative or very low pressures can lead to cavitation, air entrainment, or even pipe collapse in extreme cases.

5. Important Assumptions and Limitations

While Bernoulli’s equation is powerful, civil engineers must remember its limitations:

• Flow is assumed steady and incompressible
• Fluid is ideal (no viscosity) — head losses due to friction are neglected
• Flow is along a streamline
• No energy addition or extraction (no pumps or turbines between the two sections)

In real projects, we modify Bernoulli’s equation by adding a head loss term (\( h_L \)) and pump head (\( H_p \)) wherever applicable:

$$\frac{P_1}{\rho g} + \frac{v_1^2}{2g} + h_1 + H_p = \frac{P_2}{\rho g} + \frac{v_2^2}{2g} + h_2 + h_L$$

6. Concluding Remarks

For a practising civil engineer, Continuity and Bernoulli’s equations are not merely academic formulae. They form the backbone of hydraulic design decisions — from selecting pipe diameters to ensuring adequate pressure in distribution systems and avoiding cavitation in high-velocity zones.

A clear understanding of how velocity, pressure, and elevation interact allows engineers to design safer, more economical, and hydraulically efficient systems. Whenever a pipe changes diameter or elevation, these two equations should be the first tools applied before proceeding to more complex analysis involving friction losses or software-based modelling.

Prepared in the interest of practical understanding of fluid mechanics applications in Civil Engineering.