The real boreole is a twin-set
but for simulation, I have started
with a single cylinder
The job is made easier by thinking of the borehole as a vertical cylinder - which it is! This can't expand upwards or downwards. The aspect ratio of height to width is similar to a high rise building, in which the facade is doing most of the work - the roof and footprint are minor surface areas by comparison. If you apply the idea of Thermal Capacity to the volume, it holds several thousand kilowatt hours worth of energy, and this quantity is shrinking or growing, depending on the time of year.
The notional idea of the energy bulb volume growing and shrinking is therefore related to the increasing and shrinking radius of the theoretical cylinder - not growing cubically in all directions, but like a fattening cylinder around an axis, absorbing replacement heat only through the 'facade' of the cylinder. The solar heat that is in the earth that feeds the borehole comes from above and around the borehole, and not below.
|the contours of temperature may|
fluctuate during the day but the
natural energy is coming from the
side, not from deep below
In the case of the Peveril Solar house, it is actually two cylinders, but I had to start somewhere, with one cylinder. I can consider the Perimeter which is 2*PI*R but that is irrational as the perimeter shrinks in winter just as the delta-T is stronger. The volume of the cylinder is a function of its cross sectional area, and that is based on the Square of the Radius - PI*R^2.
I think of the pull of energy as being like something elastic. If you have a very strong spring, bungee or catapult or bowstring that is stretched, and you can assume that if it doesn't break, the pull back to the centre grows stronger as a proportion of the Square of the distance, not the Linear distance. Doubling the distance will quadruple the force. The Potential Energy in a spring is 1/2 * k * x^2 . Hooke's Law.....
|Some of the parameters in the GDL simulation model|
Practically, what does this mean? when the bulb of energy has nearly reached its summer maximum, the rate of natural charge slows to almost nothing (due to the inverse square law), but the Sunbox input might be pushing it beyond that during a hot summer. When the energy bulb is shrunken in winter, its desire to spring back is strongest. Imagine a cylinder of flexible foam rubber, with a steel rod down the axis to prevent linear extension. As this was more deeply compressed (e.g. underwater), its resistance (desire to spring back) would become strongest when it is approaching its smallest radius.
How does one equate volume here if the temperature is changing? Well imagine that we use thermal capacity equations (Energy=Thermal Capacity * delta-T) to assume that the temperature stayed always at the mean temperature of 12.8º, but the volume that would be at that temperature would increase. So in winter, when the temperature is 5º, the real world volume is the same, but we could calculate how small the energy volume would be if it was still at 12.8º. When the summer temperature is 13.5º the real world volume is the same but, we can recalculate what the energy volume would be if the temperature was only 12.8º.
By the way, it helps to assume a uniform material. Thermal capacity is Mass • Thermal capacity coefficient, so I have to assume the Clay-Marl that we have down there.
I have found that with natural recharging only, the ground in late summer seems to return to 12.8º. I have also observed that when solar charging is applied, the ground seems to maximise at just under 14º, after which one can only assume that the energy bulb is moving outwards. During the day, the ground is frequently at 14.5º or 16º or even 20º temporarily around the pipe. I always leave the measurement until midnight after a sunny day, to allow the energy bulb to mellow out for several hours after sunset.
Applying the idea
I experimented with this concept of the inverse square, and found that the result made sense. The curve looked credible. Because I am dealing with thousands of kilowatt hours, the Constant needed to be about a millionth. The user of the model needs to be able to apply a tiny amount of adjustment to this, depending on some other factors in the model, so it might be 0.00000013, so the user enters a figure of 13 and the formula applies the millionth. A small error of judgement about this variation and the borehole can shrink to nothing (because the heat pump draws more than the Sunbox puts down), and then it would be trying to find the square root of a negative number. In the other direction, it can balloon out too far, which would be obviously wrong. With some practice, one can get it nicely tuned.
The other consideration is to nominate the Pivot Energy point - the energy level that is the maximum likely in Summer. It is like a football that is at rest, at its natural size. If the bulb grows beyond this, more energy input will cause a loss. During the winter-time, this is the normal size of the energy bulb that the shrunken energy bulb wants to return to - it defines the distance that the 'spring' is pulling that radius back to summer levels.
I also have to nominate a Starting Energy level. If the model started in Winter, or if it starts in March or October, I have to make an assumption about the size of the energy bulb based on where it starts from. The dynamic simulation then continues on from that point.
More about this later. Please comment if you can shed more light on this.