Sadly not much. 100W would not effectively heat a 40 gallon water heater, let alone an 80 gallon. Also, remember that the panel likely won't really put off 100W unless your element has the exact right resistance for your specific panel.
You can do some simple calculations if you're curious how much heat that will produce...
If you use 100 Watts for 4 hours, that is 400 Wh (Watts * hours = Wh)
It takes approximately 2.4 Wh to heat 1 gallon of water 1 degree Fahrenheit.
So 400 Wh per day could heat 10 gallons of water 16.67 degrees. (The equation is [(Wh) ÷ 2.4 ÷ (#Gallons) = (change in degrees). So in other words, if your water comes into the tank at 70° (pretty warm) and you use 10 gallons per day, the theoretical temperature it could produce would be 86° (not very impressive is it?). This will somewhat reduce your draw on the on-demand but not by much.
So doing a little math the other way, if your water came into the tank at 50° and you wanted it to heat up to 120°, that is a 70° increase. If I used -say- 30 Gallons per day and I wanted to heat it up 70°...
The formula is (Wh)=(Change in Degrees)*(#Gallons)*2.4
So that would be [Wh = 70° * 30 Gal * 2.4] which comes to 5040 Wh. Spread that across 4 hours and you can see that you could do it with 1260 Watts.
I understand that your goal isn't to heat your water exclusively with the solar but the numbers show how to do the basic math. For example, if I really do use 30 Gallons per day and I heat it 70° above its original temperature, my total need would be around 5040 Wh so if I offset it by 400 Wh (100W for 4hrs) then I would be offsetting apporximately 5/63, (or 7.9%) of my usage
Please note that this doesn't account for a myriad of factors that could affect it, such as heat-loss, inefficiency, or the derating of your solar panel... These numbers should some idea but don't expect them to be exact. Please let me know if this was helpfull or if there is anything I could explane better by leaving me a comment.
Thank-you,
Maxfield Solar
This isn't one of those games where the goal is to use only this bag of parts, right?
Charge the battery with the solar panel
Simply connecting a solar panel to a battery will not do what you want. It will overcharge and wreck the battery. You need a charge controller between panel and battery. This is a keystone product that will make or break your build.
Good ones are hard to find. Generally the ones sold at chain/big-box retail stores are foreign dreck, often with reputable brand names slapped on them. I prefer using a low-end controller from a well-reputed company in the solar business like Morningstar. Unfortunately many of these are specialty products, so you just have to read up on solar-power forums and places like that to see who's regarded as the best.
From there, it's a simple affair to connect solar panel to charge controller, and charge controller to battery.
A good charge controller will not need you to throw switches or change wires.
Hook up your loads
Since your goal is lighting, get 12V lights. The point is to use 12V lighting throughout, run straight off the battery. Don't even think of running an inverter to run 120V lighting, that's just crazy.
Off-grid power is too precious to use anything but LEDs for lighting. Nothing else is efficient enough. You don't want to double your battery and solar just to run CFLs, or 7x your battery and solar to run incandescents. That's just crazy. No, LEDs are not ugly light (more on that). Yes, the government forcing swirly CFLs is unfair, but LED is also winning fair-and-square because it's better.
You can get 12V LED lighting in all sorts of shapes and sizes:
Be careful with your choice of color temperature and CRI when buying LEDs. A lot of the early or cheap LEDs had poor CRI and harsh color temperatures, and they got a bit of a reputation. These days you can get any CRI or color temperature you want.
CRI is Color Rendering Index, or how good the light looks to humans; aim for 80 or up (out of 100). Color temperature is how "blue" the light is. The traditional warm incandescent lighting is 2700-3000K (kelvin, weird unit, I know). 4100K is office fluorescent lights. 5100K is a cloudy day, blue-sky is 6500, and that sounds great when you're buying it, but it looks awful at night. Try to have all your LEDs be about the same color temperature.
Other loads
Make a very serious effort to find 12V versions of anything else you want to run. These days a lot of flatscreen TVs are 12V friendly. Boaters, RVers, tiny-house and VanLife people have good sources.
Inverting to make 120V involves a serious amount of energy loss. Even an inverter sitting idle has a fairly high power consumption.
You can't run them 24x7 or they'll destroy your battery. In fact, don't neatly install one. Leave it something you temporarily hook up in a tangled mess, so you'll remember to unhook it and put it away.
Don't overwork the battery
Both your batteries are lead-acid type. That particular type has a serious problem: it *really, really does not like being deep-cycled.** Deep-cycle lead-acids, like your Optima yellowtops, do better -- but they still will be destroyed by deep cycling, just will get more cycles before they do (on the upper range of the numbers I'm about to give).
- If you drain it dead, you'll get 5-30 cycles before battery death
- If you drain to 50%, maybe 20-200 cycles
- if only 25% DOD (using 25% of capacity) many hundreds of cycles
- 15% DOD thousands of cycles.
Factor this against the fact that lead-acid batteries typically fail anyway after 5-7 years. The upshot is, depending on your usage, you may need a larger battery for this system to make sense. Or...
Other battery types
You may think "Wait, my laptop/iPad/phone doesn't have that problem!" Correct. But before you start salvaging 18650's out of laptop batteries, lithiums have a different problem: spectacular battery fires. To avoid that, you need to build battery packs carefully and include a protection circuit - many people are doing this for off-grid / home-power. There are lots of instructionals out there, some not so safe.
There are more traditional battery types such as the famous NiFe "Edison battery" or its brothers NiCd or NiMH. But you're not going to find those on the cheap.
Best Answer
Battery Size
I would want a battery sized, as already suggested by others, roughly 2x your total expected daily output (based on summer days = more usage). That would keep you to 50% discharge on the shortest nights (but remember, you have a couple of months of those in a row) and less on the longest nights (all confusing because this is the exact reverse of the typical "charge with solar in the day and discharge at night"). In other words, calculate your total usage, multiply by the number of hours of the longest day (i.e., longest discharge time) and then double it. That is your battery size requirement. Let's say (for simplicity) it is 70W x 16 hours = 1,120 Wh x 2 = 2,240 Wh. 2,240/12 = 187 Amp-hours. So using your 100 Amp-hour 12V batteries, I would go with 2 of them.
Make sure the batteries can handle the cycles - you are talking about a significant (~ 50%) cycle EVERY DAY. Some battery chemistries can handle that a lot better than others.
Charger Size
Get a MUCH bigger battery charger. 12V 5A = 60W - that is roughly the same as your discharge rate, so at best on a 12/12 (spring/fall) day you would be breakeven. I'd want to charge at least 2x the expected discharge rate to have a fighting chance for long nights and other factors. Obviously the batteries have to be able to handle it, but if your batteries can support it then I would look for a charger that, at peak, charges on the order of 140W for your 70W usage.
AC/DC
Why convert from DC to AC if your equipment (a) can't ever use AC directly because the AC isn't available during the time you need power (which is also quite unusual) and (b) your equipment is all electronics that generally run on DC anyway. You may need some sort of regulator (perhaps as simple as a "car cigarette lighter power adapter") to use 12V DC, but that will avoid the losses of DC->AC->DC conversion, which will further reduce power usage.