I am new to the forums, so forgive me if I breach some element of protocol. I am a mentor for a team of high school students building an OpenROV for use in studying frogs in Lake Titicaca. It is a joint project between a zoo and team of biologists in Peru and Bolivia and the Denver Zoo and St Vrain Valley schools in the US. You can find our expedition described in the community. The question we want to answer is “How deep can we safely take the OpenROV in a freshwater lake at 12,500 feet?”. Our own calculations are below. We’d like to have someone else double check them, as we’d hate to have an implosion once we get down there.
The OpenROV rover is designed to dive to 100M in salt water at sea level. As such, the rover is supplied with a tether that is 100M in length. For depth, the limiting factor is the pressure differential between the ambient air pressure at the surface of the water and the pressure at depth. The pressure vessel that contains the electronics is sealed at the surface of the water (lake, ocean, etc.). That effectively sets the internal pressure of the vessel (Pa, or pressure of the ambient atmosphere). Once the vessel is submerged, the outside of the vessel is subject to the combined pressure of the atmosphere above the surface of the water PLUS the pressure due to the weight of the water column above the vessel (Pd, or pressure at depth). At sea level, Pa is one atmosphere, or 14.7psi. At 12,500 feet (altitude of Lake Titicaca), the atmospheric pressure is 9.16psi, or about 62% of the pressure at sea level. This means that the pressure inside the vessel will be significantly less at our target site than it is for the normal (sea level) usage of the OpenRov.
To calculate the pressure at depth, we need to know the density of the water. The US Navy uses different densities for fresh water and salt water. For sea water, they use 0.445psi per foot of depth. For fresh water, they use 0.433psi per foot of depth. The OpenROV design depth is 100M in salt water, or 328 feet. At that depth, the pressure due to the water (Pw) is:
Pw = 328 feet * 0.445 psi/ft = 145.96 psi
The total pressure at depth is therefore
Pd = Pa + Pw = 14.7 psi + 145.96 psi = 160.66 psi
Now the big question. What causes leaks to occur in the pressure vessel? It is (we believe) seepage of the water impinging on the O-rings. Water seeps into the gaps in the acrylic vessel until it reaches the seal made by the O-ring. The o-ring provides a mechanical connection squeezed into place by the tight-fitting acrylic end cap inserted into the cylinder of the vessel body. The design depth of the OpenROV would indicate that the O-ring can protect the contents of the pressure vessel up to a pressure differential of
Pd - Pa = Pw, or 145.96 psi.
This equation shows that the pressure differential is simply the weight of the water column at depth. In other words, the pressure differential is
Pd – Pa = (Pa + Pw) – Pa = Pw.
If this is indeed the case, for Lake Titicaca (at 12,500 feet) the reduced atmospheric pressure has essentially no physical effect on the pressure differential. To determine how deep we can go, we simply need to know how deep 145.96psi is in fresh water. That calculation is simply:
Max depth in fresh water = 145.96 psi / 0.433 psi/foot = 337 feet.
So, unexpectedly for me, I find the maximum depth to be greater in fresh water at altitude than it is for salt water at sea level. From a safety perspective, the 328 foot tether will effectively limit the rover to a safe depth.
Now, I believe the math is correct, but when I double check this number against human dive tables, I find the equivalent dive depth for humans is only 210 feet. This is directly proportional to the depth at which the pressure on the body increases by the pressure at the water’s surface (33 feet at sea level, 21 feet at 12,500 feet). I believe this has more to do with nitrogen narcosis and the lower atmospheric pressure effects of preventing the bends than it does with the simple physics of maintaining a consistent maximum pressure differential. However, we could be wrong. If we’re actually looking at a maximum pressure ratio (Pd/Pa), we get a much different answer (230 ft maximum depth). This approach, though, does not seem consistent with the physics. So, the question is, should we base our maximum depth on a consistent maximum pressure differential (as in the calculation above) or do we need to base it on a consistent pressure ratio (outside pressure to inside pressure)?
I look forward to your insights and guidance, as they are most welcome. Thanks in advance.