By my rather crude approximation, a 10mm pipe with 40 mm of insulation at 60 K above ambient would lose 1 K about every 25 seconds. Obviously at more reasonable temperatures it takes a bit longer. Still, ten minutes of cloud....

Here's my calculation, all done per metre of pipe:

10 mm O/D, 8.8 mm I/D, 61e-6 m² cross-section.
Density, 1000 kg/m² and SHC 4180 J/kg·K so linear heat capacity 255 J/m·K.

40 mm of insulation with a conductivity of 0.04 W/m·K - wild guess, probably a bit too conductive but also a bit thick. U value 0.04/0.04 = 1 W/m²·K.

Crude approximation by ignoring the fact that there's less area on the inside of the insulation than the outside: assume the area of the insulation is the area half way out, radius about 25 mm (half 40 mm plus radius of pipe) so 2·PI·r = 0.16 m²/m so heat loss is 0.16 W/m·K.

At 60 K temperature difference heat loss is 9.6 W/m.

(255 J/m·K) / (9.6 W/m) = 26.56 seconds/K

But we're only looking for rough orders of magnitude.

Anyway, supposing we did want a bit more precision then the maths is probably not too hard but using a computer is even easier:

> function tempLossRate(temp) { return 0.16/255 * temp; }
> var dT = 1.0;
> var t = 0.0;
> for (var temp = 60; temp > 0; temp -= dT) {
... console.log(temp, t);
... t += dT / tempLossRate(temp);
... }
60 0
59 26.562499999999996
58 53.575211864406775
57 81.05366014026885
56 109.01418645605833
55 137.47400788462977
54 166.4512806119025
53 195.96516950079138
52 226.0359242177725
51 256.68496267931096
50 287.93496267931096
49 319.80996267931096
48 352.3354728833926
47 385.5385978833926
46 419.4481723514777
45 454.0949114819125
44 489.5115781485792
43 525.7331690576701
42 562.7971225460423
41 600.7435511174708
40 639.615502336983
39 679.459252336983
38 720.3246369523677
37 762.2654264260518
36 805.3397507503762
35 849.6105840837096
34 895.1462983694239
33 942.0212983694239
32 990.3167529148784
31 1040.1214404148784
30 1091.5327307374591
29 1144.6577307374591
28 1199.6146272891833
27 1256.5342701463262
26 1315.562047924104
25 1376.860124847181
24 1440.610124847181
23 1507.016374847181
22 1576.3098531080504
21 1648.7530349262322
20 1724.6458920690893
19 1804.3333920690893
18 1888.2149710164576
17 1976.7566376831244
16 2070.5066376831246
15 2170.1160126831246
14 2276.3660126831246
13 2390.2052983974104
12 2512.801452243564
11 2645.613952243564
10 2790.5003158799277
9 2949.8753158799277
8 3126.958649213261
7 3326.177399213261
6 3553.8559706418328
5 3819.4809706418328
4 4138.230970641833
3 4536.668470641833
2 5067.918470641833
1 5864.793470641833
7458.543470641833

So it's down to 40 °C after 10 minutes, at 20 °C after half an hour and 6 °C after an hour.

Doesn't k = 0.16/255 = 0.000627 K/(s·K) = 0.000627 s⁻¹ ?

Posted By: JSHarrisbut the saving is so modest when compared to 15 mm pipe that I doubt it's worth worrying about unduly

It is a small effect overall, but given that (assuming that you've decided to use copper anyway) - the 10mm is both much easier to install than the 15mm and also better performing, why bother with the 15mm?

Posted By: djh
Impressive precision!

Well, yes, I did wonder if +/- 1 microsecond would be good enough.

There's a specific formula for the Y-value of pipe insulation (I think they're called Y-values when they are linear rather than U-values for areas). I expect it's on the Navi site somewhere or else google will throw it up. I do remember there are magic values involved ('critical radius').

Yes, I have seen that somewhere but was just doing a quick and dirty calculation so didn't bother to search. It's a bit magic as I recall as it only needs the ratio of the inner and outer radii, not the absolute values, as the scale cancels with the circumferences if you see what I mean.

Yep, Wikipedia says:

http://en.wikipedia.org/wiki/Pipe_insulation#Heat_flow_calculations_and_R-value

This has some significance here. 20 mm of insulation on a 10 mm pipe will be coming up to twice as effective (from the W/(m·K) point of view) as 20 mm of insulation on a 22 mm pipe.

That's quite remarkable, and most unintuitive. The maths in that paper had integrals in it which is where I always lose the plot. Why does a small amount of insulation make things worse?

Posted By: wookeyWhy does a small amount of insulation make things worse?

DJH's last link (Wikipedia) explains it very succinctly.

DJH's last link (Wikipedia) explains it very succinctly.

True, but he posted that with the caveat that it's 'partly wrong', so I wasn't sure how correct it was. So the explanation is that surface area increases faster than heat-transfer attenuation to start with. Fair enough. Now DJH needs to say what part is wrong.

Pedantic semantics aside, plastic pipe (mlcp) is used in places with as much sun as Mallorca, and has survied several seasons to my knowedge. I've had to add to such systems, and had to use the same material, to avoid ripping out and starting again. But in principle, copper or ss - depending on your ideology - neither has massive advantages - are the materials for the job.

Many panels stop at a max 180C, (30C direct sun, not your usual UK conditions) many have drainback, with oversize storage so temps can't reach stagnation levels, so use of plastic in some parts (return) of a system may be countenanced in future legislation. But to be safe here, for all classes of reader, we're better sticking to metal.