Thermal resistance of borehole heat exchangers composed of multiple loops and custom shapes
© Koenig. 2015
Received: 16 September 2014
Accepted: 29 April 2015
Published: 17 June 2015
The thermal resistance of a borehole can be reduced by employing thermally enhanced grout, increasing the surface area of the loop and locating the legs proximal to the bore wall. Thermal models that are used to predict borehole heat exchange are characterized by either simplified formulations that are restrictive in their application, but utilitarian, or complex multi-dimensional analyses that are cumbersome to implement.
The borehole resistance methodology presented here offers a straightforward solution that is built on single loop conduction shape factor analysis with thermal shunt accounting and pipe-pipe configuration analysis, to extend to multi-loop borehole configurations and custom kidney extrusions.
The borehole resistance predictions are compared to published data and information listed by manufacturers of multi-loop products in third party thermal tests against standard loops. The results are found to agree within the constraints posed by the model assumptions. The methodology offers a straightforward solution that can be incorporated into popular geothermal loop sizing software such as GLD, GLHEPRO and other system software.
The advantages and challenges of these advanced loop designs are discussed and conclusions drawn. By reducing the bore resistance, one can take advantage of less drilling and proportionally less capital cost for the bore field, while achieving the same loop temperatures.
KeywordsGeothermal heating and cooling Ground source heat pump (GSHP) Vertical loops Borehole studies Loop field Multiple loops in borehole
One of the challenges to the ground source heat pump market is the cost burden of implementing the ground heat exchange. For the case of closed-loop vertical borehole designs, the thermal barrier presented by a loop can be significant. The high-density polyethylene (HDPE) pipe loop contacts the bore wall via a thermal grout. For example, a 7–8 °C temperature difference between the water circulating in the loop and the bore wall reduces the heat pump operating performance. In general, the loop legs are not constrained within the bore cross section, and during installation, the legs can twist and contact one another over a substantial portion of the bore length, thereby degrading performance. Typically, the loop is sealed to the bore wall with a bentonite grout mixture, of which the hydraulic and thermal properties are very important to insure optimal performance of the installation. The use of a thermally enhanced grout, while advantageous to heat transfer between the loop and the bore wall, presents its own challenges. For example, the addition of sand to enhance conduction also increases thermal shunting between the legs of the loop and presents challenges in uniform mixing and deployment through a tremie pipe.
Acknowledging these challenges, a study was undertaken to understand the potential benefit of multi-loop and custom extrusion designs which offer greater surface area for heat transfer, while locating the pipe proximal to the bore wall. This borehole design strategy could reduce the capital cost of the ground heat exchanger by reducing the amount of drilling required to serve a load. The contractor or design engineer has the option of either reducing the amount of drilling, while achieving the same end-of-season temperature as a standard loop, or retaining the same well depth and achieving better heat pump seasonal performance. The first approach reduces capital cost of the loop field but retains the same annual savings, while the second does nothing to lower capital cost but rather improves annual utility savings. For example, the building owner might be more interested in lowering the loop field installation cost by 20 % and accepting less annual operating savings. Another strategic advantage might derive from less land utilization and associated mess.
The purpose of this publication is to provide an analytical basis for calculating borehole thermal resistance (BTR) for the case of multiple loops and custom extrusions. The desire is to incorporate the borehole model into existing well field sizing software packages, such as GLD, GLHEPRO, and system modeling software, such as TRNSYS (Klein et al. 1996; Shonder & Hughes 1997) and HyGCHP, on the market. To accomplish this, the thermal analysis of a single loop (Koenig and Helmke 2014) was chosen as the starting point, acknowledging the contributions of others and developing toward a practical thermal model of a borehole with multiple parallel loops.
Building on the quasi-steady-state modeling (Eskilson 1988) for vertical ground heat exchange, there have been a numerous publications on borehole resistance for a single loop. Previous studies employed finite element numerical techniques (Sharqawy 2009; Sagia 2012; Liao 2012). While these are rigorous, they have also proven to be cumbersome to construct. Others have proposed analytical solutions (Hellström 1991; Gu 1998; Philippe 2010) that are readily applied but are restricted to a limited number of pipe geometries. While these studies support the understanding that increasing the grout thermal conductivity and locating the pipe legs proximal to the bore wall enhances borehole performance, it was left to Redmund (1999) and Beier (2012) to actually demonstrate this fact in field tests. A finite element model for the case of two loops in a borehole was published (Diersch 2010), following the work of Al-Koury et al. (2005). Finally, a recent comparative study of helical pipe and triple U-tube loop designs in foundation pile (Zarrella et al. 2013) is included here, as pertinent to this study.
Borehole thermal resistance for a single loop
The temperatures, in general, can all be considered different, such that T 1 > T 2 > T b for the cooling season or T b > T 2 > T 1 for the heating season.
The total borehole thermal resistance is defined as (R p + R f)/2.
The three equations can be solved simultaneously for the heating rates using matrix methods:
Parameters chosen for this study
152 mm (6 in.)
42 mm (1 1/4 in.)
Pipe c-c spacing
64 mm (2 1/2 in.)
Pipe wall (HDPE)
Pipe thermal conductivity
0.389 W/m°C (0.225 BtuH/ft°F)
Pipe thermal resistance
0.089 m°C/W (0.154 ft°F/BtuH)
Grout thermal conductivity
0.78 W/m°C (0.45 BtuH/ft°F)
The relative amount of heat shunted between pipe is calculated from the ratio q 3/(q 1 + q 2). This turns out to be an important and useful number, because it defines what fraction of the pipe surface area is effective in heat transport to/from the bore wall. This will be apparent shortly in the shape factor analysis of individual pipe heat transfer.
For example, Table 1 lists select parametric values in a typical geo-exchange application:
For the selected parameters in Table 1, for example, the pipe to bore resistance is 0.219 m°C/W. Equation (8), however, is based on a single off-centered pipe conducting to an unobstructed surrounding wall; therefore, one might anticipate that the presence of a second pipe (leg of the loop) would contribute a similar measure, so that the total would simply be half of the single pipe contribution, as suggested by the resistance mesh model given earlier. Unfortunately, this is not quite right in that there is conductive interference due to the presence of the pipe representing the second leg of the loop.
The latter results from solving the matrix equations for q 3 in terms of the heating rate difference (q 1 − q 2).
The benefit of the closed-form quadratic solution is that it avoids the need for the T information, which requires the full bore simulation, by providing a steady-state value for R f straightaway. The two approaches give the same result, with the matrix solution offering additional information on q.
Comparison to others
Extension to multi-loop borehole geometries
In principle, adding additional loop(s) to a borehole should decrease borehole resistance by increasing the available pipe surface area for heat transfer. It should be noted, however, that the heat transfer effectiveness of an individual loop diminishes with each additional loop. This is because the heat transfer surface area of an individual loop for exchange with the bore wall is occulted by the presence of additional loops.
Two-loop borehole resistance model results
N 100p =
Avg/2N L =
As an example, Fig. 7 shows four loops (eight pipes) arranged evenly around a hollow central form of diameter, d cyl.
So, for example, with a 152-mm (6 in.) bore and a 27-mm (3/4 in.) pipe size, the maximum number of pipes that fit the constraint is ten or, equivalently, five loops. The outside diameter (OD) of this assembly is then (d cyl + 2d), or, in this example, 114 mm (4.5 in.). Other sizes are possibly limited only by the assembly size impact on insertion in the borehole and the ability to insert a grout “tremie” pipe inside the ID of the form, or, ideally, use the form itself as the grout pipe.
Multi-loop pipe-pipe spacing for shunt analysis
Configuration fraction (CF)
One of the advantages of additional loops is seen in the “N L” factor that appears in the denominator which decreases borehole resistance as loops are added. An additional benefit is that the pipe wall thickness, assuming a fixed DR for pressure rating, decreases as the pipe gets smaller to accommodate the loops.
From thermal measurements by Bowman Geothermal of REHAU 1-in. double-loop product in side-by-side installations, they report a BTR of 0.11 (PEX2), a 40 % reduction from single loop, which, after adjustment for HDPE, matches the model with an RMS error of 1.5 %.
The utility of the proposed borehole thermal model is such that it can be easily extended from simple loops to more complex multi-pipe and custom shape loop configurations. The model addresses only the thermal resistance inside the borehole (BTR), and not the diffusion component into the surrounding formation. The benefit is that it can be incorporated easily into existing software, such as GLHEPRO, to handle multi-loop borehole configurations, avoiding the details of constructing custom numerical models through FEFLOW.
The model produces reasonable results that support the data published by others. The extension of the model to multiple pipe loops in a single borehole is accomplished by (1) defining the mid-plane boundary between downcomer and upcomer pipe, (2) calculating the borehole resistance for each pipe-pipe separation across the boundary, (3) multiplying by the relative number of occurrences of each configuration to yield an average borehole resistance for a representative loop, and finally (4) dividing by the number of loops. Since the loop to bore wall temperature difference is controlled by the product, q(R b), it is noteworthy that it makes no difference whether this is treated as a single loop, in which case (q/N L)R b, or the full multi-loop design, q(R b/N L).
The benefit and practicality of implementing multiple loop borehole designs to improve performance and/or reduce installation cost is arguable. The use of a thermally enhanced grout with a single loop appears to lower borehole resistance by 30 % over neat bentonite grout at standard U-bend separations, despite higher thermal shunting between the legs of the loops. But, unless the pipe legs can be guaranteed to be separated and not twisted and contacting over a substantial length of the bore, this 30 % benefit may never be realized in practice. Physical separation, such as the use of Geo-Clips, is at least as beneficial as thermally enhanced grout. With this in mind, the obvious benefit of multiple loops in reducing borehole resistance is increased surface area for heat transfer and thinner HDPE walls that result from employing smaller diameter pipe loops. Another advantage to multiple loops is in lowering the pressure drop of the ground heat exchanger, which can be as much as 35 %. This benefit develops from dividing the flow into multiple pipe loops of a given cross section as well as shortening the length. In operation, this can save considerable pump energy over the year.
With that said, the question remains whether the assembly of pipe loops can be cost-effective with the benefit of less required drilling. On paper, the savings in drilling more than offsets the extra cost of multiple pipe assembly, which ultimately means that plastic is less expensive than the lifecycle operation of the drill rig. Certainly, there are excellent targets for commercialization of new loop technology, such as commercial applications where space for the loop field is always an issue, and the opportunity to repurpose abandoned water wells that could serve residential customers without extending the depth of the well. Ultimately, driller pricing and competition will prove the winner.
A borehole thermal model for vertical loops was developed from a modified shape factor analysis. An interference factor, f, defined by (1 − q shunt/q), the relative size of the thermal shunt between the loop segments, was employed as the modification required to properly model borehole thermal resistance. The model predictions were compared to reported results and found to match quite nicely over a range of pipe sizes and spacing. Following this success, the model was extended to multi-pipe loop geometries consisting of two-, three-, and four-loop assemblies in a single borehole and a custom kidney extrusion. The borehole resistances were calculated for each of these configurations and presented as a compilation of predicted performance. These predictions were compared to the available data reported by manufacturers of multi-pipe loop assemblies and found to give reasonable results.
- T p :
temperature of pipe loop segment, °C
- T b :
temperature of borehole perimeter, °C
- ΔT 1n :
temperature difference pipe 1 to (n = b bore; n = 2 pipe 2), °C
- q n :
exchange heating rate (pipe to bore wall or pipe to pipe), W/m
- R p :
thermal resistance of pipe (convection and pipe wall), m°C/W
- R fs :
single pipe thermal resistance (pipe to bore wall), m°C/W
- R f :
effective thermal resistance (pipe to bore wall), m°C/W
- D :
borehole diameter, m
- d :
outside diameter of the loop pipe, m
dimension ratio: pipe diameter to wall thickness
- S p :
center to center spacing between pipe, m
- h cv :
convection heat transfer coefficient, W/m2°C
- k p :
HDPE pipe thermal conductivity, W/m°C
- R s :
pipe-pipe shunt thermal resistance, m°C/W
- k g :
grout thermal conductivity, W/m°C
- d r :
diameter ratio = d/D
- S r :
spacing ratio = S p/d
- X pp :
pipe to pipe dimension = [2(S p/d)2 − 1]
- X f :
pipe to bore wall dimension = 1/2 [d r(1 − S r 2) + 1/d r]
- f :
loop pipe interference factor in thermal exchange
- d cyl :
central cylinder diameter, m
- N L :
number of loops in borehole
- R b :
borehole thermal resistance (BTR), m°C/W
- d eq :
equivalent diameter of pipe loops actively exchanging heat, m
- Z :
diagonal c-c distance between pipe in multi-loop geometry, m
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