When using GHEtool Cloud to calculate the required borehole depth, you may encounter a gradient error. In this article, we’ll explain the background of this error, why it occurs, and how you can work around it.
Calculate required borehole depth
As we discussed in our first article on this topic (which you can find hier), calculating the required borehole depth is an iterative process that starts with an initial estimated depth. The fluid temperatures are then calculated and checked to see whether they remain within the required thresholds. If they do not, the borehole depth is increased and the entire process is repeated. This is also illustrated in the graph below.
Although this iterative process should result in a borehole depth that meets the required temperature thresholds, there are some situations in which no solution exists, and an error is thrown. This is due to the temperature gradient.
Gradient error
Sizing with a constant ground temperature
Imagine that there is no temperature gradient in the ground—so, regardless of depth, the undisturbed ground temperature remains constant. In this case, the maximum average fluid temperature will always decrease as the borehole depth increases, due to the increase in total borehole length.
Since the peak injection power remains the same (as it is determined by the building demand), a greater total borehole length results in a lower specific heat injection rate per metre of borehole, which in turn leads to a lower fluid temperature. This is illustrated in the figure below.
!Hinweis
If you’re not familiar with how the specific heat injection relates to the fluid temperature, you can read our article on the short-term effect hier.
As can be seen in the graph above, the maximum average fluid temperature indeed decreases with increasing depth and converges to a constant difference from the ground temperature. This difference is caused by the effective borehole thermal resistance, which is always present in borefields. If the final fluid temperature remains below the temperature threshold, a solution can always be found.
Sizing with temperature gradient
As we discussed in our article on ground parameters (which you can find hier), the earth typically becomes warmer with depth. So, although it is still true that increasing the total borehole length decreases the specific heat injection, leading to a lower fluid temperature, there is another factor at play.
By drilling deeper, the higher average ground temperature increases the fluid temperature, resulting in a graph like the one shown below.
As you can see in the graph, both effects are now combined: there is a sharp decrease in the maximum average fluid temperature at first, followed by a (small) increase over time due to the rising ground temperature with depth. This results in a convex curve for the maximum average fluid temperature (shown as ‘Combined effect’).
If the minimum of this curve lies above the temperature threshold, then there is no mathematical solution to the sizing method, and a gradient error is thrown.
Gradient error
To illustrate this error further, let’s take a look at the figure below, which is a simplified version of the previous graph.
With two boreholes, the maximum average fluid temperature (for this example) is always above the threshold, so no solution can be found. The way to resolve this is to increase the number of boreholes, as shown in the graph on the right. Since there are now more boreholes in the borefield, the specific heat injection is lower for the same borehole depth. This causes the entire graph to shift downward, resulting in a feasible solution.
Borefields that are limited by the maximum average fluid temperature will therefore always benefit from having more, shallower boreholes rather than fewer, deeper ones.
!Hinweis
Another solution—if the maximum temperature is not a concern—is simply to increase the maximum temperature threshold, so it will no longer be a limiting factor in the design.
Numerical convergence
As should be clear from the figure above, there are two solutions that satisfy the threshold temperature, each with a different borehole length. One solution corresponds to a case where the ground temperature has little influence on the fluid temperature, and the maximum average fluid temperature is mainly determined by the specific heat injection per metre of borehole. The other solution occurs when the ground is already rather warm, but the specific heat injection is relatively low.
Since the first of these two solutions is typically cheaper—due to reduced drilling—it is, mathematically speaking, not always straightforward to predict which optimum the iterative method will converge to.
The figure above on the left shows the traditional iterative approach. First, an initial guess is made, after which a new depth is calculated. This process continues until the temperature threshold is reached (or a gradient error is thrown). Depending on your initial guess, load profile, ground parameters, borehole thermal resistance, etc., it is not known a priori which solution the iteration will converge to.
Within GHEtool, we’ve developed a novel sizing method that makes use of the underlying physics of the borefield. Instead of iterating back and forth around a certain optimum, our algorithm starts with the most shallow option and converges towards the first solution, giving you the most affordable borefield.
!Hinweis
This approach is unique to GHEtool, and a scientific paper detailing the underlying methodology and physics is currently in preparation.
Schlussfolgerung
This article explained the gradient error that can occur when calculating the required borehole depth with GHEtool Cloud. This error is caused by an increasing ground temperature and can be overcome by increasing the number of boreholes.
It should be clear that for borefields limited by heat injection, it is always more beneficial to have more, shallower boreholes rather than fewer, deeper ones.
Literaturverzeichnis
- Sehen Sie sich unsere Videoerklärung auf unserer YouTube-Seite an, indem Sie klicken hier.
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