I'm having trouble identifying a blown SMD tantalum capacitor. The markings on it are 10 35L which I took to be the capacitance, voltage rating, and tolerance, but looking up that component on the schematic provided with it, that same component is labeled with the value 47U/16V. As this capacitor blew after being subjected to a 13 VDC power supply for an extended period of time instead of the 9 VDC the board is rated at, a 16 volt rating seems more likely than 35 volts. A neighboring cap has the label 220-10L which more closely matches the schematic for it's value which is listed as 220U/6.3V. This makes me wonder if I have a different revision of the schematic than the board or if the labeling is using a different scheme. What's the value of the capacitor I need to replace it with?
Generally you are right that per marking it should be 10uF/35V capacitor. You did not state who the manufacturer is, they may simply have used gray market to source cheap remarked components.
I recommend you contacting developer/manufacturer with question about deviation in the board and circuit, not telling about voltages you applied. Because as soon as they will hear about your doubts and special conditions, they may immediately state that it is not warranty case and that it is your fault, proposing repair on T&M basis.
Hear their answer first (append info from them to your question), and then decide what to do.
P.S. If developer and manufacturer are different entities, then developer may not know what manufacturer is doing, thus raising quality issue would be a good action towards developer and your peer users of this product.
the 220-10L is a vishay email@example.comV (10V surge) the other part, the 10 35L is most likely a Chinese copy of a vishay capacitor (that might be sold as various brands, like xicon) 10uf @20V (35v surge)
You always have to keep in mind its surge and when design with these capacitors. They have a tendency to short out with inrush current. These are my least favorite caps to build with because they don't have a good longevity compared to normal electrolytic capacitors. Only suitable for throw away devices, and even then, I don't like them because most stuff that get them get thrown away and arsenic (the primary chemical in them) is poisonous to the ground water. People really need to look at the environmental impact they do when it comes to this, regardless how cheaper they are.