Bernoulli's Equation Applications for the FE Exam
Bernoulli's equation is one of the most versatile and frequently tested equations on the FE exam. Understanding how to apply it to different flow scenarios is essential. This guide covers the most common applications you will encounter.
Review: The Energy Equation
For steady, incompressible flow between two points with head losses and a pump or turbine:
P_1/gamma + V_1^2/(2g) + z_1 + h_pump = P_2/gamma + V_2^2/(2g) + z_2 + h_turbine + h_L
Where h_pump is the energy added by a pump, h_turbine is the energy extracted by a turbine, and h_L is the total head loss (major plus minor).
Application 1: Venturi Meter
A venturi meter measures flow rate by creating a pressure difference in a constriction. Applying Bernoulli between the inlet (section 1) and the throat (section 2), and combining with the continuity equation:
Q = C_d * A_2 * sqrt[2g * (P_1 - P_2)/(gamma * (1 - (A_2/A_1)^2))]
Where C_d is the discharge coefficient (accounts for real fluid effects, typically 0.95 to 0.99 for a well-designed venturi). Without friction losses, C_d = 1.
Key insight: The flow rate depends on the pressure difference, which can be measured with a differential manometer.
Application 2: Pitot Tube
A pitot tube measures fluid velocity by comparing stagnation pressure (where velocity is zero) to static pressure:
V = sqrt[2 * (P_stagnation - P_static) / rho]
This comes directly from Bernoulli's equation between a point in the flow and the stagnation point on the pitot tube. Pitot tubes are used to measure airspeed in aircraft and flow velocity in pipes and wind tunnels.
Application 3: Orifice Flow
Flow through an orifice (a hole in a wall or plate) is a common FE exam problem. For flow from a large tank through a sharp-edged orifice:
Q = C_d * A * sqrt(2 * g * h)
Where h is the height of liquid above the center of the orifice and C_d is the discharge coefficient (typically 0.60 to 0.65 for a sharp-edged orifice). This is known as Torricelli's theorem when C_d = 1.
Application 4: Tank Draining
When a tank drains through an orifice at the bottom, the flow rate decreases as the liquid level drops. The velocity at any instant is given by Torricelli's theorem:
V = sqrt(2 * g * h)
The time to drain can be found by relating the rate of volume decrease in the tank to the outflow rate and integrating.
Application 5: Flow Over Weirs
Weirs are structures used to measure or control open channel flow. For a rectangular sharp-crested weir:
Q = C_d * (2/3) * sqrt(2g) * L * H^(3/2)
Where L is the weir length and H is the head above the weir crest. For a 90-degree V-notch (triangular) weir:
Q = C_d * (8/15) * sqrt(2g) * tan(theta/2) * H^(5/2)
Application 6: Siphon Flow
A siphon allows liquid to flow upward and over a barrier, then down to a lower elevation. Apply Bernoulli between the liquid surface in the upper reservoir and the outlet:
- The liquid surface velocity is approximately zero (large reservoir).
- Both surfaces are at atmospheric pressure (or whatever the boundary conditions are).
- The driving force is the elevation difference between the upper and lower liquid surfaces.
The critical point in a siphon is the highest point, where pressure is lowest. If the pressure drops below the vapor pressure, cavitation occurs and the siphon breaks.
Application 7: Pump and Turbine Problems
When a pump adds energy to the flow, the energy equation becomes:
P_1/gamma + V_1^2/(2g) + z_1 + h_p = P_2/gamma + V_2^2/(2g) + z_2 + h_L
Where h_p is the pump head. The power required by the pump is:
Power = gamma * Q * h_p / eta
Where eta is the pump efficiency.
For turbines, the turbine extracts energy: h_t appears on the right side instead of the left. The power output is:
Power = gamma * Q * h_t * eta
Common Mistakes with Bernoulli's Equation
- Applying it where it does not apply: Bernoulli's equation requires steady, incompressible flow. Do not use it for compressible flow (high Mach number), unsteady flow, or across regions with significant viscous effects.
- Forgetting to include head losses: Real pipe systems have friction and minor losses. Include h_L when applying the energy equation to real systems.
- Incorrect datum: The elevation terms (z) must be measured from the same reference datum. Choose a convenient datum and be consistent.
- Using gauge vs. absolute pressure inconsistently: Either use gauge pressure throughout or absolute pressure throughout. Do not mix them.
- Neglecting velocity head at large surfaces: For a large tank surface, the velocity is approximately zero and V^2/(2g) can be neglected.
Problem-Solving Strategy
- Identify two points along the flow where you know conditions or want to find unknowns
- Write the energy equation between those two points
- Apply the continuity equation if there are two unknown velocities
- Substitute known values and solve for the unknown
- Check units and verify the answer is reasonable
Bernoulli's equation problems follow consistent patterns. Once you recognize the application type (venturi, orifice, siphon, pump), the setup is straightforward. Practice identifying the application and setting up the equation quickly.