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Fluid Mechanics

Rotameter (Variable-Area Flowmeter) Simulator

A tool to compute the flow rate of a "rotameter" — a float inside a tapered tube. Change the float diameter and density, the fluid density and the tube taper to see the volumetric flow, gap area and velocity update in real time, and feel how the float position maps almost linearly onto flow rate.

Parameters
Float diameter
mm
Diameter of the float inside the tapered tube
Float density
kg/m³
Float material density (aluminium ≈ 2700, stainless ≈ 8000)
Fluid density
kg/m³
Density of the measured fluid (water ≈ 998, air ≈ 1.2)
Tube taper ratio
How fast the tube radius widens with height (radius increase / height)
Float height (reading height)
mm
Height the float floats above the bottom of the tube
Float discharge coefficient Cd
Dimensionless coefficient for the float shape and flow loss
Results
—
Volumetric flow Q (L/min)
—
Annular gap area (mm²)
—
Net force on float (N)
—
Gap velocity (m/s)
—
Float height (mm)
—
Flowmeter state
—
Rotameter cross-section — flow and float

Fluid flowing upward pushes the float up, and the float settles at the height where drag + buoyancy = weight. The side scale reads off the flow rate directly.

Flow Q versus float height
Flow Q versus tube taper
Theory & Key Formulas

$$Q=C_d\,A_{ann}\sqrt{\dfrac{2g\,V_f(\rho_f-\rho)}{\rho\,A_f}}$$

Volumetric flow Q. The float rises until the flow drag balances its submerged weight (gravity - buoyancy). Because the tapered tube widens linearly, the gap area A_ann — and therefore Q — is nearly proportional to float height.

$$A_{ann}=\pi\,(r_{tube}^{2}-r_{float}^{2}), \qquad r_{tube}=r_{float}+\text{taper}\cdot h$$

Annular gap area A_ann. r_float: float radius, r_tube: tube radius at float height h, taper: taper ratio.

$$F_{net}=(\rho_f-\rho)\,V_f\,g, \qquad V_{gap}=\dfrac{Q}{A_{ann}}$$

Net force on the float F_net (gravity - buoyancy) and the velocity through the annular gap V_gap. V_f: float volume, g: gravitational acceleration.

What is a Rotameter?

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In labs and factories you see those gauges with a small ball floating inside a glass tube, and you read the flow from where the ball sits. How does that actually work?
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That is a "rotameter" — properly, a variable-area flowmeter. Look closely and the tube is narrow at the bottom and very slightly wider toward the top. A "float" sits inside. The fluid flows from bottom to top and passes through the gap between the float and the tube wall. The stronger the flow, the harder it pushes the float up, so the float settles higher. The ball's height equals the flow rate.
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I see. But if a strong flow pushes it up, wouldn't the float just shoot off the top? Why does it stop partway?
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Great question. The key is the tapered tube. The higher the float goes, the wider the tube, so the gap between float and tube (the annular ring) opens up. A wider gap lets the flow pass more easily, which weakens the upward drag on the float. So the float settles at the height where "upward drag + buoyancy" exactly balances the "downward weight". Raise the "float height" slider on the left and you will see both the gap area and the flow increase.
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So it stops at a balance point. Then is there a reason the scale is evenly spaced? The flow has a square root in it — I'd expect it to curve.
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That is the clever part of a rotameter. The float's submerged weight is constant, so what is inside the square root stays almost constant. What really sets the flow is the factor in front — the gap area. And the tapered tube is built so the radius grows nearly in proportion to height. So the gap area, and the flow, are nearly proportional to float position. The result is an evenly spaced linear scale you can read by eye, no table needed. Look at the "Flow vs float height" chart below — it is almost a straight line, right?
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It really is a clean straight line. So can I use the same scale for water and air?
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Careful here. The flow equation contains fluid density ρ as √((ρf-ρ)/ρ). A scale calibrated for water will be far off if used on an oil of different density, or on air whose density is orders of magnitude smaller. Drop the "fluid density" slider toward air, and you will see the flow change dramatically for the same float height. Gases also change density with temperature and pressure, so a density correction at the operating condition is indispensable.
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So each fluid needs its own calibration. Why is the rotameter still so widely used then?
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Because it is cheap, robust and needs no power. The structure is just a tapered tube and a float. With no electricity and no computer, you read the flow on the spot — it works even in a blackout. So it is still in active use for lab cooling water, medical oxygen, plating-bath chemical feeds — anywhere you want to confirm "is it flowing, and how many litres per minute". It is a deliberately simple instrument: accuracy is a few percent, and it must always stand vertical.

Frequently Asked Questions

A rotameter is a vertical, slightly tapered tube — narrow at the bottom and gradually wider toward the top — with a float inside. The fluid flows upward and passes through the annular gap between the float and the tube wall. As the flow rate increases, the upward drag on the float grows, and the float rises until the drag plus buoyancy balances its weight. Because the tube is wider higher up, a higher float position means a larger gap area. The height at which the float settles therefore represents the flow rate directly, so you simply read it off the scale.
The tapered tube is made so that its radius increases almost linearly with height. The annular gap area between the float and the tube depends mainly on the increase in tube radius (the float radius is fixed), so the gap area is also nearly proportional to height. Since flow rate is proportional to the gap area, flow rate ends up nearly proportional to float position, giving an evenly spaced, linear scale. This is the rotameter's greatest advantage: you can read the flow by eye, with no formula or conversion table.
Yes — be careful here. Flow depends on fluid density ρ through the term √((ρf-ρ)/ρ). Using a scale calibrated for water on a chemical of different density, or on a gas with a density orders of magnitude lower, causes large errors. For gases, density also changes with temperature and pressure, so a density correction for the operating condition is essential. When you change the fluid, recalibrate for its density and viscosity, or apply a density-correction factor when reading the scale.
Because a rotameter works by balancing against gravity, it must be installed vertically (so the float moves up and down) with flow going from bottom to top. Mounting it tilted makes the float rub against the tube wall and corrupts the reading. Pulsating flow or sudden flow changes make the float bounce up and down, so it cannot be read accurately. For gases and low-viscosity liquids a damping orifice is sometimes added. A dirty float changes its weight and shape, causing errors, so choose a design that is easy to clean.

Real-World Applications

Flow management in labs and analytical instruments: Carrier gas for gas chromatographs, cooling-water lines, the individual streams of pure-water systems — rotameters are a staple for small-flow management in research settings. Needing no power and readable on the spot, several can be lined up on an instrument's front panel so you see at a glance which stream is flowing at how many L/min. The intuitive operation of opening a valve slightly to set the float position is also popular with experimenters.

Medical oxygen delivery: The flowmeter that delivers oxygen from a cylinder or piped supply to a patient (the oxygen flowmeter) is a typical rotameter. The float height shows how many litres per minute of oxygen are flowing, letting nurses and doctors set and check it quickly. Being for gas, it is calibrated with a scale matched to the operating pressure.

Chemical feed in plating and water-treatment plants: Make-up liquid to plating baths, dosing lines for coagulants and chlorine, and similar applications use rotameters to hold a small chemical flow steady. With corrosion-resistant glass or fluoropolymer tubes and chemical-resistant floats, even aggressive chemicals can be handled. Cheaper than electronic meters and installable simply by inserting it into the pipe, the rotameter is valued for its convenience on site.

Teaching and conceptual understanding of fluid mechanics: A rotameter embodies "drag", "buoyancy", "force balance" and "the relationship between area and flow" in a single instrument, making it an excellent introductory teaching aid for fluids. By trying — as in this simulator — how the flow changes when you vary float density or fluid density, the physics of drag balancing gravity becomes hands-on understanding.

Common Misconceptions and Pitfalls

The most common misconception is that "a rotameter can be used as-is for any fluid". Fluid density ρ clearly enters the flow equation, and diverting a meter calibrated for water onto an oil or a gas makes the reading badly wrong. Gases in particular change density with temperature and pressure, so even the same "air" meter needs a correction if the operating pressure differs. When you change the fluid, the rule is to apply a density-correction factor when reading, or recalibrate for the new fluid. Note also that for highly viscous liquids the flow around the float enters a low-Reynolds-number regime and the discharge coefficient Cd itself changes.

Next, the assumption that "the mounting orientation can be slightly tilted without a problem". A rotameter works by balancing against gravity, so it must stand vertical with flow from bottom to top. Mounting it tilted makes the float rub against the tube wall, adding friction, so the reading comes out low or the float sticks. If pipe routing only allows a near-horizontal orientation, you should choose a different flowmeter type, not a rotameter.

Finally, jumping to the conclusion that "a bobbing float means a fault". Pulsating flow, or flow changes from pump start-up and sudden valve operation, make the float bounce up and down and become hard to read. This is not an instrument fault — it is a sign that the flow itself is unsteady. For a stable reading, take piping-side measures such as a sufficient length of straight pipe upstream or a buffer tank, and a damping orifice. Also, entrained air bubbles lift the float too high and cause an over-reading, so degassing deserves attention too.

How to Use

  1. Enter float diameter (mm) and material density (kg/m³)—typical glass floats range 6–16 mm at 2500 kg/m³, while stainless steel floats are denser at 7850 kg/m³
  2. Set fluid properties: density (kg/m³) for water at 1000 or oil at 860, and tube taper angle (degrees) typically 0.5–2° for industrial rotameters
  3. The simulator calculates equilibrium float height where buoyant force plus drag equals weight; read volumetric flow Q in L/min and annular gap area in mm²

Worked Example

Glass float (diameter 8 mm, density 2500 kg/m³) in water (1000 kg/m³) through a tube with 1° taper. At equilibrium height of 145 mm, the annular gap expands to 18.5 mm², generating gap velocity of 3.2 m/s and net force near zero. Volumetric flow reads 6.8 L/min. For comparison, a 10 mm stainless steel float (7850 kg/m³) in the same water circuit rises to 165 mm with flow of 12.3 L/min and gap area of 24.7 mm².

Practical Notes

  1. Rotameter accuracy degrades if float sticks to tube wall—use ball floats (spherical) in viscous fluids like glycerin (1260 kg/m³) to maintain stability
  2. Scale readings are nonlinear: doubling flow does not double float height; recalibrate if fluid density changes (e.g., switching from water to diesel at 840 kg/m³)
  3. Tube taper below 0.3° causes sluggish response; above 3° introduces measurement hysteresis; 0.8–1.2° is optimal for most process control applications