## Heating and the ground

Here's an interesting calculation to do. Imagine having solar heating panels on your roof, and, whenever the water in the panels gets above 500C, pumping the water through a large rock under your house. When a dreary grey cold month comes along, you could then use the heat in the rock to warm your house. Roughly how big a 500C rock would you need to hold enough energy to heat a house for a whole month? Let's assume we're after 24kWh per day for 30 days and that the house is at 160C. The heat capacity of granite is 0.195 x 4200J/kg/K = 820J/kg/K. The mass of granite required is:

mass energy heat capacity x temperature difference

100 000 kg,

Table E.14. Vital statistics for granite. (I use granite as an example of a typical rock.)

100 tonnes, which corresponds to a cuboid of rock of size 6m x 6m x 1m.

### Ground storage without walls

OK, we've established the size of a useful ground store. But is it difficult to keep the heat in? Would you need to surround your rock cuboid with lots of insulation? It turns out that the ground itself is a pretty good insulator. A spike of heat put down a hole in the ground will spread as

y/Antct exp

4(K/(Cp))t where k is the conductivity of the ground, C is its heat capacity, and p is its density. This describes a bell-shaped curve with width

water quartz granite earth's crust dry soil

Table E.15. Thermal conductivities. For more data see table E.18, p304.

for example, after six months (t = 1.6 x 107s), using the figures for granite (C = 0.82kJ/kg/K, p = 2500 kg/m3, k = 2.1 W/m/K), the width is 6 m.

Using the figures for water (C = 4.2kJ/kg/K, p = 1000 kg/m3, k = 0.6 W/m/K), the width is 2 m.

So if the storage region is bigger than 20 m x 20 m x 20 m then most of the heat stored will still be there in six months time (because 20 m is significantly bigger than 6 m and 2 m).

Limits of ground-source heat pumps

The low thermal conductivity of the ground is a double-edged sword. Thanks to low conductivity, the ground holds heat well for a long time. But on the other hand, low conductivity means that it's not easy to shove heat in and out of the ground rapidly. We now explore how the conductivity of the ground limits the use of ground-source heat pumps.

Consider a neighbourhood with quite a high population density. Can everyone use ground-source heat pumps, without using active summer replenishment (as discussed on p152)? The concern is that if we all sucked heat from the ground at the same time, we might freeze the ground solid. I'm going to address this question by two calculations. First, I'll work out the natural flux of energy in and out of the ground in summer and winter.

temperature (°C)

temperature (°C)

Feb Mar Apr May Jun

Aug Sep Oct Nov Dec

Feb Mar Apr May Jun

Aug Sep Oct Nov Dec

Figure E.16. The temperature in Cambridge, 2006, and a cartoon, which says the temperature is the sum of an annual sinusoidal variation between 3 °C and 20 °C, and a daily sinusoidal variation with range up to 10.3 °C. The average temperature is 11.5°C.

If the flux we want to suck out of the ground in winter is much bigger than these natural fluxes then we know that our sucking is going to significantly alter ground temperatures, and may thus not be feasible. For this calculation, I'll assume the ground just below the surface is held, by the combined influence of sun, air, cloud, and night sky, at a temperature that varies slowly up and down during the year (figure E.16).

### Response to external temperature variations

Working out how the temperature inside the ground responds, and what the flux in or out is, requires some advanced mathematics, which I've cordoned off in box E.19 (p306).

The payoff from this calculation is a rather beautiful diagram (figure E.17) that shows how the temperature varies in time at each depth. This diagram shows the answer for any material in terms of the characteristic length-scale z0 (equation (E.7)), which depends on the conductivity k and heat capacity CV of the material, and on the frequency w of the external temperature variations. (We can choose to look at either daily and yearly variations using the same theory.) At a depth of 2z0, the variations in temperature are one seventh of those at the surface, and lag them by about one third of a cycle (figure E.17). At a depth of 3z0, the variations in temperature are one twentieth of those at the surface, and lag them by half a cycle.

For the case of daily variations and solid granite, the characteristic length-scale is z0 = 0.16 m. (So 32 cm of rock is the thickness you need to ride out external daily temperature oscillations.) For yearly variations and solid granite, the characteristic length-scale is z0 = 3 m.

Let's focus on annual variations and discuss a few other materials. Characteristic length-scales for various materials are in the third column of table E.18. For damp sandy soils or concrete, the characteristic length-scale z0 is similar to that of granite - about 2.6 m. In dry or peaty soils, the length-scale z0 is shorter - about 1.3 m. That's perhaps good news because it means you don't have to dig so deep to find ground with a stable temperature. But it's also coupled with some bad news: the natural fluxes are smaller in dry soils.

The natural flux varies during the year and has a peak value (equation (E.9)) that is smaller, the smaller the conductivity.

For the case of solid granite, the peak flux is 8W/m2. For dry soils, the peak flux ranges from 0.7 W/m2 to 2.3 W/m2. For damp soils, the peak flux ranges from 3 W/m2 to 8 W/m2.

What does this mean? I suggest we take a flux in the middle of these numbers, 5 W/m2, as a useful benchmark, giving guidance about what sort of power we could expect to extract, per unit area, with a ground-source heat pump. If we suck a flux significantly smaller than 5 W/m2, the perturbation we introduce to the natural flows will be small. If on the depth 1

depth 2

 fP Vt^ % || — 8 Vv\ \\> J )\ \13 | \ \ 10 \ \ I12 11 12 J

Jan Mar May Jul Sep Nov Jan depth 0

depth 1

depth 2

depth 3

### Jan Mar May Jul Sep Nov Jan

Figure E.17. Temperature (in 0C) versus depth and time. The depths are given in units of the characteristic depth z0, which for granite and annual variations is 3 m. At "depth 2" (6 m), the temperature is always about 11 or 12 0C. At "depth 1" (3 m), it wobbles between 8 and 150C.

other hand we try to suck a flux bigger than 5 W/m2, we should expect that we'll be shifting the temperature of the ground significantly away from its natural value, and such fluxes may be impossible to demand.

The population density of a typical English suburb corresponds to 160 m2 per person (rows of semi-detached houses with about 400 m2 per house, including pavements and streets). At this density of residential area, we can deduce that a ballpark limit for heat pump power delivery is

This is uncomfortably close to the sort of power we would like to deliver in winter-time: it's plausible that our peak winter-time demand for hot air and hot water, in an old house like mine, might be 40 kWh/d per person.

This calculation suggests that in a typical suburban area, not everyone can use ground-source heat pumps, unless they are careful to actively dump heat back into the ground during the summer.

Let's do a second calculation, working out how much power we could steadily suck from a ground loop at a depth of h = 2 m. Let's assume that we'll allow ourselves to suck the temperature at the ground loop down to AT = 5 0C below the average ground temperature at the surface, and let's assume that the surface temperature is constant. We can then deduce the heat flux from the surface. Assuming a conductivity of 1.2W/m/K

 thermal heat length-scale flux conductivity capacity k CV Z0 A\/CyKU> (W/m/K) (MJ/m3/K) (m) (W/m2) Air 0.02 0.0012 Water 0.57 4.18 1.2 5.7 Solid granite 2.1 2.3 3.0 8.1 Concrete 1.28 1.94 2.6 5.8 Sandy soil dry 0.30 1.28 1.5 2.3 50% saturated 1.80 2.12 2.9 7.2 100% saturated 2.20 2.96 2.7 9.5 Clay soil dry 0.25 1.42 1.3 2.2 50% saturated 1.18 2.25 2.3 6.0 100% saturated 1.58 3.10 2.3 8.2 Peat soil dry 0.06 0.58 1.0 0.7 50% saturated 0.29 2.31 1.1 3.0 100% saturated 0.50 4.02 1.1 5.3

Table E.18. Thermal conductivity and heat capacity of various materials and soil types, and the deduced length-scale Zq = y and peak flux A^CyKco associated with annual temperature variations with amplitude A = 8.3 0C. The sandy and clay soils have porosity 0.4; the peat soil has porosity 0.8.

(typical of damp clay soil),

If, as above, we assume a population density corresponding to 160 m2 per person, then the maximum power per person deliverable by ground-source heat pumps, if everyone in a neighbourhood has them, is 480 W, which is 12 kWh/d per person.

So again we come to the conclusion that in a typical suburban area composed of poorly insulated houses like mine, not everyone can use ground-source heat pumps, unless they are careful to actively dump heat back into the ground during the summer. And in cities with higher population density, ground-source heat pumps are unlikely to be viable.

I therefore suggest air-source heat pumps are the best heating choice for most people.