Sears DieHard 27582

Sears sells the DieHard model 27582 marine (deep cycle) battery for $130 (excluding core deposit). This is probably the cheapest chemical (Sealed Lead Acid or SLA) battery in its size range on the market in terms of initial cost ($ per Peukert capacity).  The 27582′s total power output is not obvious, given that SLA discharge efficiency varies depending on instantaneous output current, significantly affecting the runtime between charges.

The Peukert capacity C is defined as the total amp-hour output of a lead-acid battery discharged at one amp (about 12W). In order to predict the cumulative current output at higher instantaneous rates, you also need the Peukert constant k for a specific battery model. This constant varies from 1.1 to 1.6. Given k, the runtime is t(i)=C/i^k, where i is the instantaneous current output. [I am ignoring thermal effects.] Cumulative current output is therefore i*t(i)=i*C/i^k=C*i^(1-k). For an (unrealistically constant) potential of 12V, total power output P(p)=12V*i*t(i)=12V*C*i^(1-k)=12V*C*(p/12V)^(1-k), where p is the (constant) instantaneous output power.

Like many SLAs, the only two pieces of data published for the 27582 are the manufacturer’s advertised “reserve capacity” — t=(25A), to 10.5V — and “cold cranking amps” (CCA) — t(675A), to 7.2V. You might think that one could determine k by solving these two t(i) equations simultaneously. In practice, the CCA rating is a poor representation of actual capacity. The 30 second runtime drains the SLA’s surface charge rather than utilizing the battery’s entire stored energy. For instance, if you solve for the 27582′s published data, you get k=1.81, which is too high even for a low-end SLA.

In the absence of sufficient data, I’m going to brazenly assume k=1.40 for the 27582. This is not an experimentally determined constant; I would be delighted if someone can acquire a (new) 27582 or similar SLA and a few days to perform the necessary runtime measurements.  Using the reserve capacity number, C=(i^k)t(i)=(25^1.4)(200 minutes)=302Ah. In that case, the cumulative energy delivered P(p)=12V*302A*(p/12V)^(1-1.4)=3.624kW*(p/12V)^(-0.4). [Graphed below for up to 1100W output power.] A 12W draw delivers the full Peukert capacity of 3.62kWh.  At 100W, the total energy delivered is reduced by 57% to 1.56kWh. At a full kilowatt, the SLA has to deliver 83A, and the cumulative energy output drops to 618Wh.

SLA Output Energy

As a negative-exponential curve, the energy loss per change in output power decreases with increased total power. For instance, increasing output power from 100W to 200W causes a 370Wh loss in output power, but another 100W increase (to 300W total) results in a loss of only 180Wh. Conversely, placing three SLAs in parallel (or series) reduces the instantaneous current draw from each, shifting the curve to the right.  For a kilowatt draw, three 27582s must deliver only 333W each, resulting in 959Wh delivered by each SLA — a 55% improvement! Note that the total (charger) energy input required is the same for the two scenarios, so this 55% advantage applies to both energy costs and runtime between charges. Moreover, the ratio of cumulative powers is not related to the instantaneous power output. Therefore, replacing a single SLA with

• two SLAs will always increase efficiency by (1/2)^(1-k)-1, or 32% for the DieHard 27582;
• three SLAs will always increase efficiency by (1/3)^(1-k)-1, or 55% for the 27582;
• four SLAs will always increase efficiency by (1/4)^(1-k)-1, or 74% for the 27582.

This is what automotive battery manufacturers are doing when they produce a “starter” battery, designed to deliver brief but enormous currents to starter motors. They incorporate a profusion of thin lead plates into each battery, effectively creating a set of parallel SLAs within a single case. Unfortunately, these thin plates erode easily, which restricts the charge capacity of a starter battery to the extent that they become useless for long-term power delivery. They are more akin to gigantic capacitors than chemical batteries.

Naturally, the same efficiency improvement could be achieved with a lower k. AGM SLAs have a lower Peukert constant, for instance, but are at least twice as expensive for the same Peukert capacity as a non-AGM SLA. To deliver the same output energy capacity per dollar as two 27582 batteries operating at a kilowatt power output, an AGM’s k would have to be small enough such that (p/12V)^(1-k)=2(p/12V)^(1-1.4), or k=1.24. such a Peukert constant reduction of 0.16 is not a small achievement. Of course, two 27582s will weigh quite a bit more than one AGM. For immobile installations, this is not an issue, which is why solar power enthusiasts tend to store their energy in many inexpensive SLAs (with high Peukert constants) wired in parallel rather than a few AGMs.

The graph calculation spreadsheet is available for download [OpenOffice ODS format]. The various battery parameters can be changed to analyze total output power and cost per kWh delivered.