How deep can we go at altitude?


#1

Hi, Community!

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.

Craig


#2

Welcome to OpenROV!

Your project sounds extremely interesting and I look forward to more updates about your progress.

Your pressure calculations look correct and operation in fresh water usually allows for a greater depth, but I have some suggestions. In general these figures are very theoretical. The actual maximum depth that an OpenROV can resist heavily depends on a variety of factors, such as built quality, lubricant and sealing quality, temperature of the air inside the tubes, quality of acryl and many more. Also you have to look at a variety of different failures at depth. This can be leaks, but as you wrote also implosion. These occur at different depths for the different tubes. So overall it is quite hard to define a depth at which the ROV fails.

The official 100m depth stat is a safe calculations that allows for safe operation under normal conditions. However, some ROVs have been taken significantly deeper. If you use an OpenROV that is correctly built, you should be fine with your plan.


#3

Just echoing Fe3C’s comment. Your math looks pretty good to me but there are some other factors that need to be considered. As an easy-to-remember rule of thumb, it just so happens to work out that in metric units, the pressure from altitude change (in air) and depth change (in water) are both very close to multiples of ten in relation to pressure: 1m of depth = ~10 kPa (~0.1 atm) of pressure increase and 1000m of altitude = 10 kPa (~0.1atm) of pressure decrease (when at non-extreme altitudes). In other words, you can expect the equivalent of about 1m less water pressure for every 1000m altitude you’ve gained. 12500ft (almost 4000m) will result in water pressure equivalent to being about 4m shallower at sea level… BUT… if you’ve opened or equalized your vehicle while at this altitude (which I suggest you do in order to keep the o-ring seated in the right direction so it doesn’t leak when you pass through equalization at 4m depth), all bets are off. All of these calculations assume gauge pressure, not absolute pressure. Without the original 100kPa atmospheric pressure inside (at Lake Titicaca, the ambient pressure should be around 63kPa), gauge pressure at true depth goes back to the same value.

The other points Fe3C made were also very good. It turns out that the temperature of the water is also a major factor in performance. I don’t have empirical data for the tubes in v2.8 (which are thicker), but when I was validating the design of the 2-series, I did a series of tests to check performance of some weaker tubes we had in ice water in preparation for expeditions going to Antarctica. The spread is fairly wide, but as you can see, the statistical difference in failure pressure between room temperature and freezing water is about 12 psi or 83 kPa. A warm tube will have noticeably less strength than a cold one.

Long story short, if you want to go deep, the best situation is to be in cold, fresh water with a carefully built ROV.

Good luck with your expedition and please keep us posted with how it goes!

~Eric


#4

I can share a pressure related experience from today. I was diving at a height of 700m (see this post for a full report) and for some reason (excitement?) I forgot everything we talked about in this thread.

I can confirm that a tube closed at sea level, will eventually pop open, if heated from inside, even when under water (very dangerous!). I was lucky enough to quickly recover the ROV before significant leaks happened. Secondly, the IMU pressure sensors is based on sea level pressure, so it has a measurement error. In line with Eric’s argumentation there is an offset of about (1 meter depth measurement)/(1000 meter height to sea level). This might be useful for high altitude dives.


#5

If you go under the Diagnostics tab there is a setting to zero the depth. This should be done at the beginning of the dive to ensure that you have an accurate baseline.