I receive a lot of donated inner tubes, which may be perfectly good. How can I check them with confidence and be assured of no slow leak, without damaging the tube or taking an extremely long time?
I inspect tubes by following Mattnz's and Paul H's answers, If no obvious leak is found, then I proceed to visual and tactile assessment:
Most Butyl tubes in good state will be dark gray and not black. And they should not feel tacky or sticky. Also look for porous texture or crackled areas. If there are black or very dark areas and they feel tacky, sticky or porous then discard or recycle them.
Another sign of good tubes is that they should expand more or less uniformly (except where previous patching as been done). Some tubes that have already degraded or are too old, will bulge predominantly in one or two spots. (like a 4 inch diameter in a single spot in a tube that barely expands to 1.5 inch everywhere else). If that happens, it's likely that part of the tube is already weakened and prone to "spontaneous puncture".
Finally, gently move the valve stem around. It should hold on if you're not to aggressive with the movement. Some old tubes may become weak around the valve neck and they will leak a the junction between rubber and metal.
I have a stash of old tubes lying around, and that's the type of test I use when deciding to put one on an active bike. The ones that do not pass the check get the valve stem removed (to definitively disable them for tire use) and get tossed in a scrap bin. Later they get used as rubber shims or anywhere a rubber layer may be useful.
Here’s what I would do when testing patched tubes:
Inflate the tube outside of a tire just a little past the point where it starts to expand. Let sit over a night or two. If it’s still got air, you’re probably good to go.
If you want more instant feedback, fill up a bathtub or large bin with water, submerge a slightly-inflated tube and look for bubbles.
Fill to nearly twice its normal size (refer https://www.parktool.com/en-int/blog/repair-help/inner-tube-repair). A small hole becomes a larger hole, and the air leaks are faster and more obvious. Quick check is to use ears and lips to scan of obvious puncture. If none, leave over night or longer. If found scan for more. More and a couple, discard, otherwise fix. If flat after an overnight wait, use soapy water, discard if puncture not found.
Use vulcanizing patches and glue for a longterm fix. The no-glue patches are great for 'get you home' but not as reliable.
Despite the two excellent answers, I would be concerned with the uncertain age of the donated tubes, and the mechanical weaknesses introduced by old rubber losing elasticity. An old tube may look identical to a new tube, pump up just fine with no leaks and hold its inflation overnight. Put it inside a tire and have it experience the stresses of its natural habitat, however, and it may fail sooner rather than later.
I carry extra tubes in my seat bag and I put them separately in a plastic bag to retain the elasticizing compounds in the rubber, as well as to provide a bit of protection from the other items in the bag. I chuck them in the rubber recycling bin after 2 years.
After losing control and crashing from an old tube in the front wheel of my bike suddenly losing integrity, perhaps I'm a bit paranoid, but in the long run, better safe than sorry.