Single or double U-tube? Part 2: Hydraulic aspects

The debate between the single and double U-tube has many different viewpoints. Last time, we discussed this question from a thermal perspective, and in this week’s article, we will focus on the hydraulic side of things: the pressure drop.

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If you have not read the first article in this series, you can find it hier.

Single or double? That is the question

In the world of geothermal design, few topics seem as sensitive or as likely to spark debate as the question of using a single or a double U-tube. Last week, when we started to unravel this question from the thermal side, looking at the effective borehole thermal resistance for both situations, we found there was no definitive answer. Depending on the flow rate, the percentage of antifreeze, or even the grout thermal conductivity, either a single or a double U-tube could be preferable.

The thermal perspective, however, is only one of the different viewpoints. In this article, we will focus on the hydraulic aspects, namely the pressure drop of these different solutions, since this is linked to both the cost associated with pump sizing and the electricity consumption of the pump.

Recap: What is the pressure drop?

If you recall our article on pressure drop (which you can find hier), there is a distinction between local (or minor) losses, caused by bends, T-junctions, manifold inlets and so on, and friction (or major) losses, which result from the interaction between the fluid and the pipe wall and within the fluid itself. It is this latter aspect that is important for our discussion today.

These friction losses can be calculated using the well-known Darcy–Weisbach formula:

$\Delta P = f \cdot \frac{L}{D} \cdot \frac{\rho v^2}{2}$

where:

• $f$ the Darcy-Weisbach friction factor (-)
• $L$ the pipe length (m)
• $D$ the pipe diameter (m)
• $\rho$ the fluid density (kg/m³)
• $v$ the fluid velocity (m/s)

To make the formula above a little more insightful, we can replace the fluid velocity $v$ with $\dot{V}/A$, where $\dot{V}$ is the flow rate through the pipe (m³/s) and $A$ is the cross-sectional area of the pipe (m²), which equals $\pi D^2/4$ for a circular pipe. The Darcy–Weisbach equation can therefore be rewritten as:

$\Delta P = f \cdot L \cdot \frac{8\rho \dot{V}^2}{\pi^2 D^5}$

When we write the equation in this way, two important relationships become apparent:

1. $\Delta P \propto \dot{V}^2$
2. $\Delta P \propto D^{-5}$

The first proportionality tells us that when the flow rate doubles, the pressure drop quadruples. The second one is even more extreme: when the inner pipe diameter halves, the pressure drop across the pipe increases by a factor of 32.

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For laminar flow, the Darcy–Weisbach friction factor is $64/Re$, and since the Reynolds number is also a function of $v$ and $D$, we know that $Re \propto vD$. Given the relationship between the flow velocity and the pipe diameter described above, we can rewrite this as $Re \propto D^{-1}$. Therefore, the second proportionality, in the case of laminar flow, is $\Delta P \propto D^{-4}$.

With these two insights, we can now try to answer the question: “Which one is better, a single or a double U-tube?” from the hydraulic perspective.

Hydraulic aspects

To answer this question from the hydraulic perspective, we will consider three different scenarios:

• The same pipe diameter but a different number of pipes (that is, our original question: single or double)

• Different diameters but the same number of pipes (intermediate insights)

• Different pipe diameters and a different number of pipes (our original question revisited)

We will conclude this chapter with some nuance on pump energy in the case of modulating circulation pumps.

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Unless stated otherwise, a DN32 pipe, a borehole diameter of 140mm with a length of 100m and a grout with a thermal conductivity of 1.5 W/(mK) are assumed in this article.

Same diameter, different number of pipes

When the pipe diameter is the same, a double U-tube configuration will always be beneficial in terms of pressure drop. Since the total flow rate per borehole is now divided over two tubes, the flow rate per tube in the single U-tube configuration is twice as high as in the double one. As we saw earlier, this twofold increase in flow rate results in a fourfold increase in pressure drop, which is clearly visible below.

When we compare this graph to its thermal counterpart below (as discussed in detail in our previous article), we can see that there are regions where the double U-tube performs better from both a thermal and a hydraulic perspective. The range, roughly between 0.28 and 0.45 l/s, where the single U-tube is more efficient from a thermal perspective, comes at the cost of a higher pressure drop.

Different diameter, same number of pipes

When we take a side tour and compare two single U-tubes with different diameters, we can clearly see that the single DN40 outperforms the DN23 case, showing a lower pressure drop. This is because, with a larger internal diameter, the pressure drop (as discussed earlier) decreases to the power of five.

Just as in the subsection above, there is an overlapping range in which the single DN40 performs better on both the thermal and hydraulic fronts. The region where the single DN32 U-tube outperforms the single DN40 from a thermal perspective again comes at the cost of a higher pressure drop.

Different diameter and number of pipes

Given the insights above, let us revisit our single versus double U-tube discussion, this time with different pipe diameters. Below, you can see the pressure drop when comparing a single DN40 with a double DN32 U-tube. It is clear that there is a range, between 0.1 and 0.25 l/s, in which the single DN40 has a lower pressure drop than its double DN32 counterpart. Although this might seem counterintuitive at first (since the flow velocity is indeed higher in the DN40 case), the larger pipe diameter results in a smaller contact area between the pipe wall and the fluid, leading to an overall lower pressure drop in this laminar range.

Interestingly, when we compare this again with the thermal side of the story, we can see that there is no longer any overlapping region where the single DN40 outperforms on both the thermal and hydraulic fronts. On the other hand, for flow ranges above 0.45 l/s, the double DN32 is better from both the thermal and hydraulic side.

Pump energy and modulating pumps

Before we conclude our hydraulic discussion, let us briefly consider why the pressure drop matters. The first reason concerns pump sizing, since the pump must be able to deliver the required head to overcome the pressure drop at the design flow rate. A higher pressure drop therefore requires a larger pump and a slightly higher investment cost, although this is relatively small compared with the total cost of the geothermal borefield.

The second reason relates to pump energy use. When there is a higher pressure drop to overcome, electricity consumption also increases, leading to higher operational costs. This effect is illustrated in the table below.

For the given flow rate of 0.3 l/s, it is clear that the single DN32 has a significantly higher pressure drop and therefore also a higher yearly electricity consumption compared with the other two options. This can be explained by the higher Reynolds number, which already indicates a transitional flow regime. As seen earlier, the single DN40 also has a lower electricity consumption than the double DN32.

Historically, that would have been the end of the story. However, since modern heat pumps are increasingly modulating, the flow rate through the borehole is no longer constant. This means that if 0.3 l/s is our design flow rate (at which the pump should be selected), most of the time the actual flow rate will be around 70% of that, i.e. 0.21 l/s. This is shown in the table below.

Here we can see that the actual electricity consumption is lower in all cases, but most notably for the single DN32. It still shows the highest value, yet it is already more acceptable than at the design flow rate. This also places the importance of pump energy consumption in a broader perspective when working with modulating pumps.

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A varying flow rate also affects the thermal behaviour of the probes through the effective borehole thermal resistance. This, however, is a topic for another article. Stay tuned!

Conclusie

In this article, we examined the hydraulic aspects of the single versus double U-tube debate. Based on several comparisons, we can conclude that:

1. when comparing single and double configurations with the same diameter, the double U-tube will always have a lower pressure drop, and

2. when comparing single (or double) U-tubes with different diameters, the larger diameter will always result in a lower pressure drop.

The situation becomes more complex when both the pipe diameter and the number of pipes vary, or when thermal aspects are taken into account. In such cases, no general conclusions can be drawn, and it is better to simulate the specific situation. Furthermore, when working with modulating pumps, the importance of pump energy consumption should also be viewed with some nuance.

Stay tuned for our next article, where we will shed light on innovative probes and practical aspects related to this topic!

Referenties

• Bekijk onze video over dit artikel op onze YouTube pagina hier.