Sink Rate and Sinking Time of the Titanic

 How long will it take the Titanic to sink after she has been damaged? We know that, in reality, the iceberg was struck at 11:40 pm, and the ship sank around 2:20 am, an elapsed time of 2 hours 40 minutes (160 min). To estimate the sink rate and time with physics, we will need to estimate the time taken for the 5 compartments to completely fill with water. This represents a volume of 15625 m3.

We start with the Bernoulli equation, which relates, for two points, the speed of fluid flow, the heights of the liquids, and the pressures. The Bernoulli equation only applies for a non-viscous fluid. The general form of the Bernoulli Equation is Bernoulli equation, where P1 and P2 are the pressures at the two points, v1 and v2 are the velocities of the fluid at the two points, h1 and h2 are the heights of the fluid, and ρ is the density of the fluid. See the diagram below.

Diagram of the Titanic flooding
Water flooding a compartment of the Titanic.

We assume that the water has zero speed at the surface and that the hole in the hull is 1 m below the surface of the water. The pressure at the surface of the water and inside the hull is also assumed to be the same. With P1 and P2 being equal and with v1 = 0, simplified Bernoulli equation

Dividing by ρ and rearranging in terms of the speed which water flows into the ship, v2, we have a result known as Torricelli’s theorem, Torricelli's theorem

Using g = 9.8 m/s^2 and (h1-h2) = 1 m, we find that v2 ≈ 4.4 m/s.

We can now use the volume flow rate equation (Giambattista et al., 2006) to find the rate at which the water entered the ship. The general form of the equation is volume flow rate equation, where ΔV is the volume of a fluid flowing in the time interval Δt, A is the area through which the fluid is flowing, and v is the speed of flow.

Assuming a speed of 4.4 m/s and that the hole's area is 1 m2, we have specific volume flow rate equation, meaning that 4.4 m3 of water will enter the ship per second.

Diagram of the Titanic flooding more
Water flooding a compartment of the Titanic up to the level of the hole.

Unfortunately, it is much more difficult to determine how long the Titanic needed to sink. There are two problems. One is that the rate of water coming into the ship will change as the level of the water inside the ship approaches and exceeds the level of the hole. At this point, the water rushing in will have to “push” away the pre-existing water. (The water will continue to enter the ship until the waterline is the at the same level both inside and outside.) It is also difficult to determine when the ship has an overall density exceeding water.

However, we can calculate the flooding time for the original five compartments. We will use the volume of the watertight compartments up to the bulkheads, a volume of 20 100 m3. (This is slightly larger than the volume of the part of the watertight compartments below the water, as the bulkheads extend above the waterline.)

Using our figure for the rate of the water, we find that it will take 20 100 ÷ 4.4 = 4568 s for the five compartments to fill up to the waterline. This is approximately 76 minutes. Compare this to the actual time it took the Titanic to sink, 160 minutes. Even this crude estimate is within a factor of 3 of the actual value.

Using our box model for the Titanic, we can also estimate the total volume of the Titanic as 268.9 m × 28.2 m ×  30 m = 227 000 m3. (We are estimating the distance from the bottom of the keel to the top of the hull to be 30 m.) Using this volume, we find that the Titanic would take more than ten times longer than the time indicated above to sink – approximately 760 minutes or over 12 hours. This figure is, unfortuately, less accurate than the one above, as many ships could arrive in 12 hours and save all the passengers of the Titanic.