Bernoulli's Theorem — Pipe Flow Visualizer

The core principle is conservation of mechanical energy along a streamline for an ideal (inviscid, incompressible) fluid. The total head, or energy per unit weight, is constant.

P is static pressure (Pa), ρ is fluid density (kg/m³), v is flow velocity (m/s), g is gravity (9.81 m/s²), and z is elevation (m). Each term is an energy density: Pressure, Kinetic, and Potential.

This equation is solved together with the continuity equation, which enforces mass conservation for a steady, incompressible flow.

A is cross-sectional area (m²) and D is diameter (m). This shows the inverse relationship: velocity increases when the pipe constricts. In the simulator, changing D₁ or D₂ directly alters v₂ via this relationship before Bernoulli's equation balances the pressure.

Bernoulli's theorem states that energy is conserved along a streamline: the sum of the pressure, kinetic, and potential energy is constant.

$P + \dfrac{1}{2}\rho v^2 + \rho g z = \text{constant}$

Dividing by $\rho g$ expresses it in units of length (head).

$\underbrace{\dfrac{P}{\rho g}}_{\text{pressure head}} + \underbrace{\dfrac{v^2}{2g}}_{\text{velocity head}} + \underbrace{z}_{\text{elevation head}} = H\ (\text{total head})$

$P$ is static pressure, $v$ velocity, $z$ height, $\rho$ density, and $g$ gravity. It quantitatively explains the counter-intuitive fact that pressure drops where the flow speeds up. This simulator visualizes how these three heads are distributed along the pipe in real time.

Continuity (mass conservation): for incompressible flow the product of area $A$ and velocity $v$ (the flow rate $Q$) is constant.

$Q = A_1 v_1 = A_2 v_2, \qquad A=\dfrac{\pi D^2}{4} \;\Rightarrow\; v_2 = v_1\left(\dfrac{D_1}{D_2}\right)^2$

Where the pipe narrows the velocity rises, and by Bernoulli the pressure falls — the Venturi effect, used in flow meters, atomizers, and carburetors. The pressure components are:

A Pitot tube measures velocity $v=\sqrt{2(P_0-P)/\rho}$ from the difference between total and static pressure.

Bernoulli's theorem is idealized and holds under these assumptions:

• Incompressible: constant density (gases too, for Mach number below ~0.3).
• Inviscid: friction losses neglected.
• Steady: no change with time.
• Along a streamline (or irrotational flow).

Real pipes have a viscous head loss $h_L$ (friction and minor losses), so the energy equation is extended:

$\dfrac{P_1}{\rho g}+\dfrac{v_1^2}{2g}+z_1 = \dfrac{P_2}{\rho g}+\dfrac{v_2^2}{2g}+z_2 + h_L$

$h_L$ is the sum of pipe friction (Darcy–Weisbach) and local losses at bends and valves. For long pipes or viscous fluids $h_L$ dominates, so loss evaluation matters as much as Bernoulli's equation itself.

Venturi Meters & Orifice Plates: These are flow measurement devices that use a constriction to create a predictable pressure drop. By measuring P₁ and P₂ across the narrow throat (simulated by reducing D₂), engineers use Bernoulli's equation to calculate the flow rate. It's a direct application of the trade-off you see in the simulator.

HVAC System Design: Designing ductwork for heating and ventilation requires understanding how pressure changes with duct size and airflow speed. Engineers use Bernoulli's principle to ensure air is delivered with sufficient pressure to all vents, balancing duct diameters and fan power.

Aerodynamic Lift (Simplified): While the full story is complex, Bernoulli's principle explains part of how wings work. Air moving faster over the curved top surface has lower pressure than the slower air underneath, creating net upward force—a direct pressure-velocity trade-off.

Pump and Turbine Sizing: The "head" a pump must provide is calculated from Bernoulli's equation, considering the required pressure increase, velocity change, and elevation gain (Δz) in the system. Conversely, turbines extract energy, reducing the total head from inlet to outlet.

When playing with this simulator, you might be tempted to think of Bernoulli's theorem as a "magic law." The first key point to grasp is that it relies on three major assumptions: inviscid, incompressible, and steady flow. For instance, water is nearly incompressible, but when air flows at high speeds, compressibility effects can no longer be ignored. In practical piping design, you should cultivate the habit of constantly questioning whether these assumptions hold true for your situation.

Next, do not confuse the relationship between inlet velocity v₁ and flow rate Q. While the simulator lets you directly change v₁, in the field, it's typically the pump or fan that determines the "flow rate." For example, for the same flow rate of 10 m³/h, simply changing the inlet diameter from 50mm to 100mm causes the inlet velocity v₁ to plummet from about 1.41 m/s to about 0.35 m/s. If you don't keep this "constant flow rate" in mind while adjusting parameters, you'll create a disconnect from reality.

Finally, consider the importance of pressure measurement location. The simulator clearly shows pressures at "section 1, 2," but in an actual venturi meter, the positions of the "static pressure taps" before and after the constriction are strictly defined. Otherwise, turbulence can cause measurement fluctuations. Don't naively convert a large ΔP (P₁-P₂) from a tool directly into a flow rate, or you'll get burned. Use this tool strictly for "understanding the principle" of ideal behavior.