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Assuming a typical lead-acid, 12 V car battery (typically at 13 V or so fully charged), and that it takes roughly 500 A over 3 seconds to start an engine, how long will it take to recharge the battery at any given charge rate?

Here's my attempt from what I remember about physics:

12.8 V * 500 A = 6400 W

Over 3 seconds that's 19,200 joules.

So, in a perfect world where all the current goes right back in to the battery and whatnot, how long does it take to regain all my joules and put them back in my battery?

Given a 2A charge rate:

14 V (output of charger?) * 2 A = 28 watts

Here's where I'm a little shaky. What's next? Divide the joules by the wattage to get time? Seems like it:

19,200 joules / 28 watts = 11.4 minutes.

That's it? 11.4 minutes at 2 A and all 19,200 joules are back? Seems hard to believe. My charger also has a 10A setting. So that means in about 2.5 minutes, it'll be "recharged".

So, are my assumptions correct? Do you really just use the charging voltage to calculate this, it seems like you would need to put the charging voltage in relation to the battery's capacity/voltage/whatever.

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14v*2A = 28 Watts, but that would only be true if your battery was at 0v. If your battery was at 12v, there's only 2v difference, 2V * 2A = 4W = A very long time to recharge it. Hence why automotive alternators typically run to 100A output. –  John U Feb 11 '13 at 11:28
Ah, ok, that was another thing I was unsure about. Thanks. –  Nick Feb 12 '13 at 14:32
It also assumes your charger would manage 2A into a 12v battery, it may in reality be much less as the battery voltage comes up towards 14v. –  John U Feb 12 '13 at 15:13
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3 Answers

up vote 8 down vote accepted

"Close enough" you can use

Tcharge = Tdischarge x i_discharge / i_charge x k

k is a current efficiency factor and varies with battery chemistry, charge and discharge rates, battery state of charge and phase of the moon (and sometimes whether today is a bank holiday). BUT
for lead acid k is typically about 1.1 to 1.2
for lithium ion k is about 1.01
for NimH k is about 1.15 - 1.2

This just says that charge and discharge times are inversely proportional to current drain multiplied by a variable-constant.

The "constant" varies because of many factors. Lithium chemistries have no secondary reactions that "eat up" current input. NimH (and NiCd) have secondary chemical reactions that make gases, heat and other fun stuff and consume some of the supplied energy.

Note carefully that CURRENT ratios are NOT the same as energy charge ratios. When charging, internal resistance will DROP voltage so Vin must be greater than battery_proper as the current drop across internal resistances is lost.
When discharging, internal resistance again DROP voltage but now Vout will be lower than V_battery+proper due to internal drops. So, you lose both ways. Overall energy efficinecy = current efficiency (from above) x Vout_mean/Vin_mean.
At high currents (such as a car cranking a starter motor) up to about half the total voltage MAY be dropped across internal resistances. eg a less than fully charged, less than good condition 12V car battery may measure 6V at the battery terminals during cranking. The same battery will require up to 13.6V when charging. So voltage efficiency if discharged by cranking and charged wne the battery is almost fully charged = 6/13.6 = ~ 44%. This is after the 90% efficincy mentioned above for lead acid. SO eg a near full charged lead acid battery that is "a bit tired" may manage 0.9 x 0.44 =~ 40% energy efficiency discharged energy / charge energy.

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Wait, are you saying that LiIon and NiMH are "over-unity"?!? –  Dave Tweed Feb 11 '13 at 0:23
@DaveTweed - No - I'm saying that I had charge and discharge threads in mind and mixed them. (I usually think in terms of "how much current do I get out for 1 A in" - he is asking in terms of "how much must I put in to get xxx out". I spotted this during editing and wondered if anyone would pick up on it before I corrected it in the final version. You did :-). –  Russell McMahon Feb 11 '13 at 0:31
+1 For bringing in phase of the moon and bank holidays. I always make sure to bring those in when talking about carburetors with people. –  Nick Mar 25 at 3:03
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No, it isn't Joules in = Joules out. To a first approximation, it's Coulombs in = Coulombs out. It's the electrons flowing through the circuit that participate in the chemical reaction inside the battery (but not at 100% efficiency).

Forget about the energy/power/voltage calculation and just make the ampere-seconds for charging equal to the ampere-seconds for discharging, and then multiply by a fudge factor to account for the inefficiencies.

500A × 3s =1500 A-s = 2A × 750s = 10A × 150s

750s = 12.5 minutes

Figure about 90% efficiency, so the 12.5 minutes / 0.90 = about 14 minutes.

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Ok that's much simpler. How does the voltage play into this? Specifically, what happens if I charge at 15 V instead of 13V? How does one pick a charge voltage and how does it affect the charging? –  Nick Feb 12 '13 at 14:36
@Nick: The charge voltage simply influences the instantaneous current going into the battery, based on the battery's current internal voltage and internal resistance. Higher voltage implies more current, but the current may need to be limited to a certain value based on the physical construction of the battery. –  Dave Tweed Feb 12 '13 at 14:47
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Even if your battery delivers 500A, that's the PEAK current, when the motor is stopped and there is no Back EMF, so basically the motor is a small resistance and inductance. After the motor stars spinning, the back EMF lowers the current drained for the battery. I guess these huge currents are drained for ms only.

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Don't guess. In fact, the starter motor for an automobile engine is operating close to stall current for the duration of the cranking, until the engine catches and removes the mechanical load from the starter. It really does require a few hp (= a few kW) to crank a modern high-compression engine. However, keep in mind that the automobile's charging system, with a capacity of 50 to 100A or more, can restore the cranking charge within a minute or two. –  Dave Tweed Feb 12 '13 at 14:50
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