How do I apply Bernoulli's equation to a tank or nozzle problem?

Bernoulli's equation is one of the most common fluid mechanics tools, but students lose marks when they use it without stating assumptions. Before calculating, identify the two points, confirm they lie along a reasonable streamline, and decide which terms can be neglected. For a tank and nozzle, the usual points are the free surface in the tank and the nozzle exit.

In its head form, Bernoulli's equation is often written as:

textp1/(rho g) + v1^2/(2g) + z1 = p2/(rho g) + v2^2/(2g) + z2

where pressure head, velocity head and elevation head are all in metres of fluid.

What the question is asking

The question is asking you to apply conservation of mechanical energy to fluid flow. You need to show the physical meaning of each term, not only insert numbers. A strong answer states assumptions such as steady flow, incompressible fluid, negligible viscosity or loss, and points on the same streamline.

Key concepts to cover

• Pressure head: p / rho g
• Velocity head: v squared / 2g
• Elevation head: z
• Gauge pressure versus atmospheric pressure
• Free surface velocity in a large tank
• Exit velocity from a nozzle
• Assumptions and head losses
• Units and sense check

Worked tank example

Question: A large open tank contains water. The free surface is 2.5 m above a small nozzle outlet. Estimate the exit velocity, neglecting losses.

Choose point 1 at the free surface and point 2 at the nozzle exit.

For an open tank and open nozzle, both points are exposed to atmospheric pressure, so pressure terms cancel if gauge pressure is used. The tank is large, so the free surface velocity is approximately zero. Let the nozzle exit elevation be z2 = 0, so z1 = 2.5 m.

text0 + 0 + 2.5 = 0 + v2^2/(2g) + 0
v2 = sqrt(2g x 2.5)
v2 = sqrt(2 x 9.81 x 2.5)
v2 = 7.0 m/s approximately

The answer is reasonable because more height gives more velocity, and the units are metres per second.

Sample questions and short answers

1. Why do atmospheric pressure terms cancel?

If both the tank surface and the nozzle exit are open to atmosphere, they have the same atmospheric pressure. Using gauge pressure, both are zero, so the pressure head terms cancel.

2. When can I ignore velocity at the tank surface?

You can usually ignore it when the tank cross-sectional area is much larger than the nozzle area. The same flow rate passes through both areas, so the large tank area makes the surface velocity very small.

3. What if the problem gives head loss?

Add the head loss term to the outlet side:

textp1/(rho g) + v1^2/(2g) + z1 = p2/(rho g) + v2^2/(2g) + z2 + hL

Head loss reduces the exit velocity.

4. What assumptions should I write?

State steady, incompressible flow; negligible viscous losses unless given; points on the same streamline; and no pump or turbine work between the two points.

Common student mistakes

• Forgetting to state assumptions
• Mixing pressure in pascals with head in metres
• Using absolute pressure at one point and gauge pressure at another
• Keeping the tank surface velocity when the tank is clearly large
• Applying Bernoulli across a pump, bend, or long pipe without considering added energy or losses
• Forgetting units

How to make the answer stronger

Draw a simple diagram and label point 1, point 2, z1, z2 and the datum. Mark pressures as atmospheric if both points are open. This reduces algebra mistakes and shows the marker that your equation came from the physical system.

You can also mention limitations. Real nozzles have losses, contraction and turbulence, so measured velocity may be lower than the ideal Bernoulli estimate. In more advanced problems, use a discharge coefficient or head-loss term.

Related questions

• How do I solve a first law of thermodynamics problem for a closed system?
• How do I find the reactions of a simply supported beam with a point load?
• How do I solve a circuit using Kirchhoff's voltage and current laws?

Academic use note

This guide is for engineering assignment support. Use the method to understand your own problem, and always follow the assumptions, loss model and sign convention given by your lecturer.

Sources and further reading

• Engineering LibreTexts: Bernoulli Equation
• Geosciences LibreTexts: The Bernoulli Equation