My calculation, probably missing something, but here's what I did:
$$
1 \mathrm{\ \mu A} + (50 \mathrm{\ mA} \times 0.1\%) + 25 \mathrm{\ \mu A} =
76 \mathrm{\ \mu A}
$$
$$
\frac{76 \mathrm{\ \mu A}}{ 80 \, \% \mbox{ efficiency}} = 88 \mathrm{\ \mu A}
$$
Round up to \$100 \mathrm{\ \mu A} = 0.1 \mathrm{\ mA}\$
$$
\frac{2700 \mathrm{\ mAh}}{ 0.1 \mathrm{\ mA}} \approx 3 \mbox{ years}
$$
If you're using rechargeable batteries, they'll discharge on their own long before that. Or if any of your other calculations are off (like maybe it's a 98% instead of 99.9% sleep), that will affect it a lot too.
Something like the venerable UC2906 datasheet here will do what you want. It is specified as having a 40 Volt upper limit but this can be overcome with relative ease. The output switch control can very easily adapted to drive a higher voltage external device and the high side current sense can be and input voltage sensing can be referenced down. Annoying but doable. One probably viable approach is to float the whole charger IC at say 24 volts above ground and scale and offset the voltage sensing inputs appropriately and it would probably work quite well. This effort is potentially worthwile because the IC implements a range of lead acid battery relevant algorithms not otherwise probably easily obtained in off the shelf IC.s Doing it yourself with a microcontroller would b "relatively easy" [tm] and thy explain the algorithms well enough to allow emulation.
You need to provide more detail as to what you are doing and why. 48V usually implies something special and 2W charge is exceptionally low for a 48V LA system. Lead Acid batteries need some special care with voltage profiles and a large battery with too small charger may not be able to be properly managed.
The use of a pulse charger is best kept for areas where results do not matter or you are experimenting. The pulse charger circuit you show will charge batteries but whether it is a good idea for your battery is not knowable without more information.
Added:
I see that there are a number of devices that may be what you have.
Products page
Workhorse Monitor ADCP
Workhorse Sentinel ADCP
Others ...
Sentinel is bigger than monitor physically. Both say 20-50V external power.
@swinchen - how many of these are there?
SLA are cheap and self discharge over 1+ years is bearable with the right brand - and given the stated capacity you probably intend to solar charge them incrementally.
As they say you can use 20V - 50V V external power in, and 28-42 for internal battery, you could safely [tm] use 2 x 12V SLA for external power feed and 3 x 12V SLA for internal power. The 36V puts you inside the direct control tange of a number of SLA charge ICs.
But How do eg LiFePO4 or LiIon or even NimH compare? If you have good volume then you can get custom NimH at any capacity you want from 800 - 2500 mAh and at 800 mAh even AAA would do. However, that many cells in series poses its own challenges and is usually best avoided.
20V = say 8 x LiFePO4 but you can buy made up batteries at various voltages and capacities off the shelf. Or say 7 x LiIon as LiPo or other. LiFePO4 is good at high temperature end and bearable down to somewhat under 0 C.
Another possiblity is a continuous running converter from a battery of your choosing. Efficiencies can be 85%-90% under load and idle power can be minimal with a suitable design. This allows eg 12V SLA or one or two cell LiIon or LiFePO4 or ... . It is most likely that your PV panels are 12V nominal, and ~= 18V Vmpp - yes?. Converter noise MAY be an issue - when a linear regulator would potentially help, but an adequately quiet boost converter should be doable. If they use a buck conveter (or boost or ...) to allow 20-50V then it shows that properly designed converter noise need not be an issue.
Sentinel:
They say 20-50VDC external and 450 Watt.hour capacity at 0C for internal battery. Quite a high capacity battery. Say 10 Ah at 45V. A standard 12V 7Ah SLA brick is nominal 84 Ah capacity so that is about 6 of those !!!. That makes the 450 Wh sound like a typo. Whole unit is 300mm tall. 200mm dia.
Best Answer
It can be confusing when a product contains a battery and a voltage-changing inverter, as it's not always obvious which specs apply to the battery, and which the terminals.
This product has an internal nominally 50v battery, and a bi-directional inverter, through which charge and discharge takes place. It's designed to interface with rooftop panels, which typically have system outputs in the 350-550v range, allowing low loss connection through small wires from remote roofs.
The maximum charge power is 5kW. This is the only charging limit specified. This is consistent with the charge being limited by a battery current, yet delivered through voltage interface at 350-550v. 5kW at a 50v battery happens to be 100A. However, if the panels were outputting 500v, this would be 10A at the powerwall/panel interface. As this product is supposed to be easy for homeowners to install, I think we can assume that its internal controls take care of charge management, when hooked up to a set of roof panels, and that we 'don't need to know' what's happening at the battery level.
Similarly, the calculation for battery Ah looks plausible (I've not checked it for accuracy or even order of magnitude), but again it's irrelevant for the user, they interface with the 13.5kWh energy capacity specification at the 500v terminals. I doubt that the battery is 50v, this is a nominal, lithium batteries have a wide voltage range over their charge/discharge cycle.
The 14.3A and 20A current specifications refer to the high voltage terminals, not the battery.
I read the data as saying 'stack up to 9' units regardless of whether their AC or DC. The limitation might be purely software/communication hardware, like how many ports the controllers have. It might be only one unit connects to the house and all the others pass-thru, so there's a current limit on some connections. It might be that to stack them they go through a combining unit, with a current limitation. It might be they all connect in parallel to the house so there is no current limitation, but there might be control stability issues, connect too many in parallel and one might fight the others. They might have qualification for only a certain energy density in domestic premises, without needing higher levels of fire-proof construction round them. There is so much software, design and regulatory stuff in these products that physics is but one minor limitation.