# PV, ETs and Flatties

## An Eccentric Anomaly: Ed Davies's Blog

Following on from my previous PV For Space Heating post, it's interesting to consider what combinations of photovoltaic (PV, electric), evacuated-tube (ET) and flat-plate (thermal) solar panels make sense, particularly for the provision of space heating and domestic hot water (DHW) for any significant proportion of a winter not close to the tropics. First, a look at the performance of these panel types.

# GNUPlot

The graphs here are produced using gnuplot. If you want to play at home, here's the complete script: thermal-pv.plot. Use it with something like:

```gnuplot < thermal-pv.plot
```

to produce a bunch of PNG files in the current directory.

# PV

A typical, and pretty cheap, polycrystalline PV panel suitable for off-grid use with an appropriate (MPPT) charge controller is the Kinve KV230-60P. From the datasheet (downloaded from Solar System Warehouse) the output for varying levels of sunlight is:

Insolation (W/m²)Output (W)
1000230.2
800183.3
600138.1
40088.7
20041.7

Fitting these numbers roughly by eye to get round numbers gives the following gnuplot equation to model this panel:

```qpv(G) = (G - 40) / 4.1
```

where `G` is the insolation in watts per square metre and `qpv` is the output of the panel in watts. Note that that 4.1 might seem to imply an efficiency of nearly 25% but the panel is larger than one square metre (about 1.6 m²) so the actual efficiency is around 14%. Actually, of course, the output of an individual panel is not terribly interesting; it's pretty arbitrary what size panels you bought in the first place. In some cases, where space is limited, the efficiency of the panels, in W/m² for some amount of sunlight, is relevant. Usually, though, it's the financial cost of an individual watt of output that matters: pence/watt.

Today's Solar System Warehouse price for the KV230-60P is £170.20 (plus VAT, I assume) so the cost in p/W is:

ppv(G) = 17020/qpv(G) Obviously, energy is more expensive when it's not so sunny.

# Solar Thermal

The description of PV above is a bit plodding to illustrate the steps carefully. The same sort of logic applies to solar thermal (warm/hot water) panels with one additional complication: the hotter they are run the more heat they lose (i.e., the less efficiently they operate).

(Actually, PV suffers from a similar effect; it loses some efficiency as the temperature increases. However, this effect is small and not under our control so it's ignored here.)

The usual way to model a solar-thermal panel is to consider it as absorbing a certain, fixed, proportion of the incident sunlight then losing a certain amount of energy proportional to the difference between the average collector fluid temperature and the surroundings and a little bit more energy proportional to the square of that difference.

The usual symbols used are η0 for the zero-temperature-difference efficiency, α1 for the simple coefficient of temperature loss and α2 for the loss proportional to temperature squared. Disappointingly, gnuplot doesn't do Greek so the following will have to do instead:

```q(G, t, A, n0, a1, a2) = A * (G*n0 - a1*t - a2*t*t)
```

where `G` is, again, the insolation in W/m², `t` is the average temperature difference between the fluid in the panel and the surroundings (K or °C), `A` is the area of the panel (m²) and `q` is the output of the panel in watts.

Some typical solar thermal panels are available from Navitron. The SFB20-47 is an evacuated-tube panel with 20 47 mm diameter, 1500 mm long tubes feeding a copper insulated manifold at the top. There's a datasheet available from SPF. The price today is £391.30 plus VAT.

Similarly, the FKA-240-V is a flat-plate collector for £525.41 today, with this datasheet.

There are a couple of oddities in the way the SPF reports are presented. Firstly, they give the η0, α1 and α2 parameters three times, once for each of the gross, aperture and absorber areas of the panels. I don't think it matters which you use so long as you're consistent. I've used the gross area numbers on the grounds that the others are matters for the panel designer and not of much interest to mere users.

Secondly, they sometimes use the "reduced" temperature difference, the temperature difference between the collector fluid and ambient divided by the insolation, or something. This is just confusing in my opinion, particularly as you wind up with an extra G in the T2 term, so I tend to ignore it.

Anyway, the numbers:

  SFB20-47 FKA-240-V Evacuated Tube Flat Plate 2.403 2.514 m² 0.408 0.694 1.17 3.2 W·m⁻²·K⁻¹ 0.0029 0.0086 W·m⁻²·K⁻²

Or in gnuplot speak:

```qet(G, t) = q(G, t, 2.403, 0.408, 1.17, 0.0029)
qfp(G, t) = q(G, t, 2.514, 0.694, 3.2, 0.0086)
```

Notice how the numbers for the flat-plate collector are quite a lot larger. In other words, the flat-plate will be much more efficient at absorbing heat when the fluid temperature is close to ambient but will lose that heat more quickly as the temperature difference increases.

We can see the general shape of the curves for the two panel types by plotting their outputs assuming insolation of 1000 W/m² and 400 W/m²:

```plot [t=0:120] [0:2000] qfp(1000, t), qet(1000, t), qfp(400, t), qet(400, t)
``` This is good for seeing the shape of the curves but a bit misleading if you try to compare the panels. The flat-plate used here is slightly larger than the evacuated-tube panel and 34% more expensive. Plotting the cost of a watt for the same circumstances:

```pet(G, t) = 39130/qet(G, t)
pfp(G, t) = 52541/qfp(G, t)
```
```plot [t=0:120] [0:500] pfp(1000, t), pet(1000, t), pfp(400, t), pet(400, t)
``` Under favourable conditions (high insolation, low temperature difference) the flat-plate beats the ET on price, though not by a huge margin. However, when things start getting difficult (low insolation, high temperature difference) the ET keeps working for quite a while when the flat plate has just stopped doing anything useful at all.

# PV vs Solar Thermal

To get useful comparisons of PV against the solar thermal panels it's better to pick a few temperature differences then plot the price of power against the level of insolation. A temperature difference of 0 °C is interesting to represent the case where the solar thermal is run at low temperatures to provide tepid water as input to a heat pump. 25 °C above ambient might be useful for underfloor heating (UFH) and 45 °C for DHW when the outside temperature is around freezing.

```plot [G=0:1000] [0:500] ppv(G), pfp(G, 45), pfp(G, 25), pfp(G, 0), pet(G, 45), pet(G, 25), pet(G, 0)
``` At high insolation there's not a lot of difference between the panels. PV is quite a bit more expensive, but that's hardly a surprise. Let's expand the scale a bit to look at the performance on less bright days:

```plot [G=0:600] [0:500] ppv(G), pfp(G, 45), pfp(G, 25), pfp(G, 0), pet(G, 45), pet(G, 25), pet(G, 0)
``` The big surprise, at least to me, is that below a bit under 400 W/m² the PV starts to be cheaper than the flat panel for 45 °C water and below 300 W/m² it even beats evacuated tubes for the same temperature difference. Down nearer 200 W/m² it's even getting into the same range as flat-plate for space heating temperatures (25 °C) though ET is still noticeably cheaper.

What this is saying is that if you want to heat some water to 45 °C in weak sunshine then using a PV panel and an immersion heater could actually be cheaper than using a solar thermal collector. This is contrary to what many might tell you - probably because thinking has not been updated as PV prices have dropped.

Of course, once you allow yourself very low temperature differences the solar thermal panels win handsomely, if not immediately very usefully.