Not looking for exact numbers; just a quick-and-dirty back-of-the-napkin method for calculating the theoretical endurance of an AUV. This would be used to make decisions concerning thruster count, battery size, and vehicle dimensions.

Assume (what I would assume is the most basic form factor with which I could do this math) a torpedo-style vehicle with a perfect hemisphere for the dome, a single rear thruster, and two rotary and vertical thrusters recessed. Assume the T200 thruster. Assume that it is neutrally buoyant.

Given a selection of the vehicle’s length, diameter, and mass, how do I determine the thrust required to move it at a desired speed? Blue Robotics has the thrust-power curves for their T200 thrusters, so I would be able to quickly calculate the vehicle speed, power consumption, and endurance. I’m just trying to figure out how to calculate that thrust requirement.

Vehicle motion is governed by applied forces, and the maximum velocity at a given thrust occurs when the thrust force is equal to the drag force in the opposite direction.

F_{thrust} = F_{drag}

The thrust force (F_{thrust}) can be roughly taken from the T200 performance specs, noting that the achievable thrust is affected by thruster placement and vehicle shape (which can affect the water flowing into the thruster(s)), and thruster orientation (e.g. if the thruster isn’t pointing in the motion direction, then that needs to be factored in \rightarrow F_{thrust-forwards} = F_{thrust} \cdot \cos(\theta), where \theta is the thruster angle away from forwards).

The drag force (F_{drag}) is determined by the drag equation:

F_{drag} = \frac{1}{2}\rho v^2C_DA

where \rho is the water density you’ll be operating in, v is the flow velocity of the vehicle relative the the water (including any currents), A is the vehicle cross-sectional area in the flow direction (A=\pi \frac{d^2}{4} given dome diameter d when the flow is directly towards the vehicle front, i.e. the vehicle is moving directly forwards and isn’t going through sideways current), and C_D is the vehicle’s drag coefficient in the direction of motion.

Drag coefficient is the most difficult to determine accurately, but this textbook reference says an ROV generally has a drag coefficient between 0.8 and 1 (excluding tether drag, which presumably you won’t have for an AUV) - hopefully that’s at least sufficient for rough calculations (although those aren’t numbers I’ve verified).

where the vehicle is assumed to have a flow direction that’s directly forwards.

Further Considerations

Safety

It’s worth noting that T200s and the control electronics aren’t designed/intended to be run at maximum thrust for very extended periods, particularly if you’re running at the maximum voltage (they can end up overheating and become damaged).

Depending on the number of thrusters in active use, batteries can also overheat if the discharge current is too high for too long. Batteries generally have a burst current rating, as well as a continuous current rating, and as an example from our battery product pages:

It’s generally possible to measure battery temperature (and probably recommended if you’re using automated control), but it’s good to know about as a potential source of the system “maxing out”.

More Accurate Estimation

For increased calculation accuracy it may be important to consider the effects of battery discharge (reducing voltage) over time, and to take drag measurements instead of making rough assumptions about the coefficient.

The initial speed estimate above is calculated from a steady-state perspective, but in reality the vehicle will take some amount of time to accelerate to that velocity from stationary. If the transient time to change speeds becomes important you’ll have F_{thrust} - F_{drag} = m_{vehicle} \cdot a, but because drag depends on velocity that requires an integration/simulation to compute over time (that’s not particularly difficult to set up, just something to be aware of).

As a general note, posts with the vehicle-design tag may be worth having a look through.

Hopefully that gets you off to a decent start - feel free to post follow-ups on your progress, or any further questions you might have

Nice write-up, Eliot! Thank you for always being thorough in your replies.

One note I would make is that a torpedo-shaped AUV should be able to have a drag coefficient of much less than 1.0… perhaps closer to 0.2 or even 0.1. In fact, the thrusters and other “protuberances” will probably end up generating more drag than the body of the AUV.

Good to have a more accurate rough estimate - thanks!
I haven’t done much work with measuring vehicle drag coefficients, so while I suspected 0.8-1.0 was overly conservative it was a range I could at least provide a source for, and didn’t require making and running a CFD simulation just to get a number that had some kind of reasoning behind it. Experience definitely wins out for getting estimates that are otherwise challenging/slow to obtain