How an hourglass keeps time
Water in a clepsydra runs faster when the vessel is full and slower as it empties, because the pressure depends on the height of the water. Sand behaves differently. Grains jam against each other and against the walls, forming arches that carry most of the weight, so only the grains right above the opening push on it. The result is a nearly constant flow rate, regardless of how much sand is in the top bulb. That is why a sand glass is such a simple and trustworthy timer.
The Beverloo equation
Engineers describe the flow of grains through an opening with the Beverloo equation. It says that the mass flow rate depends on the density of the packed sand, the gravity constant and the opening diameter raised to the power of 2.5, minus a correction of about one and a half grain diameters that accounts for the empty zone near the edge of the opening, where grains cannot fit.
W = flow rate (kg/s), ρ = bulk density (kg/m³), g = 9.81 m/s²
D = opening diameter, d = grain diameter, C ≈ 0.55–0.65, k ≈ 1.4–1.6
Because of the 2.5 power, the neck is the most important dimension of the hourglass. Making the opening 10% wider increases the flow by about 27%. This is also why the opening must be cut or drilled very accurately.
From flow rate to sand weight
Once the flow rate is known, the sand you need equals the flow rate times the time. Dividing by the bulk density gives the volume, and dividing the volume by the cross-section of the bulb gives the height of the sand column in the top bulb.
Volume = mass ÷ bulk density
Height = volume ÷ (π × (bulb diameter ÷ 2)²)
How big should the bulbs be?
The bulbs must hold the sand with space to spare. A piece of sand forms a cone in the lower bulb, and the upper bulb has a funnel-shaped part where sand cannot be used efficiently. As a rule of thumb, each bulb should hold about 2.5 to 3 times the sand volume. Too small and the sand cannot flow evenly; too large and the hourglass is bulkier than it has to be.
Grain size and the neck
The opening must be several times larger than a grain, or the sand will jam and stop. Designers keep the ratio of neck to grain at five or higher, and even ten or more for sand with an irregular shape. Fine, round, very dry grains flow best. Moisture makes the grains stick together, so sealed glass and dry sand, often baked beforehand, are essential for a reliable timer.
| Typical use | Time | Neck | Sand |
|---|---|---|---|
| Tooth-brushing timer | 2 min | ~1.1 mm | ~6 g |
| Egg timer | 3 min | ~1.2 mm | ~12 g |
| Tea timer | 5 min | ~1.2 mm | ~20 g |
| Meditation timer | 15 min | ~1.5 mm | ~120 g |
| Decorative one-hour glass | 60 min | ~1.5 mm | ~490 g |
How accurate is an hourglass?
A good hourglass is accurate to within about 2–5%, which is a few seconds in a three-minute timer. Humidity, static electricity, temperature and the tilt of the glass change the flow slightly. Calibrate your timer by testing it several times, and adjust the amount of sand a little: since time is proportional to the mass, removing 1% of the sand shortens the time by 1%.