Ground Source Heat Pump Heat Transfer: Earth Loop Design and Soil Conductivity

Introduction

Ground Source Heat Pumps (GSHPs), also known as geothermal heat pumps, represent one of the most energy-efficient and sustainable HVAC solutions available. Their ability to leverage stable subterranean temperatures for heating and cooling dramatically reduces fossil fuel dependency and operational costs. Central to the GSHP system's success is the design and functionality of the earth loop—the buried piping system that acts as the thermal interface between the heat pump and the surrounding soil.

This deep dive explores the critical role of heat transfer within the ground loop, focusing specifically on earth loop design principles and the impact of soil thermal conductivity on system performance. Comprehensive coverage includes theoretical backgrounds, core calculations, equipment sizing, standards compliance, troubleshooting, and cost efficiency considerations. As HVAC engineers and designers aim to optimize GSHP systems for diverse climates and soil conditions, mastery of these elements ensures robust and reliable installations.

Technical Background: Heat Transfer in GSHP Earth Loops

The earth loop operates as a conduction heat exchanger, transferring heat between circulating fluid (usually water or water-antifreeze mixtures) and the surrounding soil mass. The effectiveness of this transfer depends primarily on the thermal conductivity of the soil, the configuration and length of the loop, and the fluid flow parameters.

Core Equations and Formulas

The fundamental heat transfer process for the earth loop can be modeled using Fourier's Law of Heat Conduction in cylindrical coordinates for borehole heat exchangers (BHEs), coupled with fluid heat balance equations.

Fourier’s Law (steady state):

q = -kA(dT/dx)

where:
q = heat transfer rate (W)
k = thermal conductivity of medium (W/m·K)
A = cross-sectional area normal to heat flow (m²)
dT/dx = temperature gradient (K/m)

For earth loop design, the heat transfer process can be further characterized by the thermal resistance model, capturing resistances from the fluid, pipe wall, grout, and soil.

Overall Thermal Resistance (R_total):

R_total = R_fluid + R_pipe + R_grout + R_soil

Where each resistance is calculated as:

Resistance Component	Formula	Units	Description
Fluid Convective Resistance (Rfluid)	1 / (hf · Ai)	K/W	Heat convection between fluid and internal pipe surface, where hf is convective heat transfer coefficient
Pipe Wall Conductive Resistance (Rpipe)	ln(ro/ri) / (2·π·kpipe·L)	K/W	Conduction through pipe wall thickness (ri inner radius, ro outer radius, L length)
Grout Conductive Resistance (Rgrout)	ln(rgrout/ro) / (2·π·kgrout·L)	K/W	Conduction through grout filling annulus between pipe and borehole wall
Soil Conductive Resistance (Rsoil)	ln(rborehole/rgrout) / (2·π·ksoil·L)	K/W	Conduction through soil surrounding borehole (rborehole radius of borehole)

Numerical Data: Typical Thermal Conductivities

Table 1: Typical Thermal Conductivity Values for Common Soil and Materials

Material	Thermal Conductivity (W/m·K)	Notes
Dry Sand	0.15 - 0.25	Highly variable with moisture content
Moist Clay	1.2 - 1.8	Higher moisture increases conductivity
Gravel	1.5 - 2.0	Good conductivity due to higher density and moisture
Bentonite Grout	0.6 - 1.2	Used for borehole fill, improves thermal contact
PEX Piping (Closed Loop)	0.35	Typical pipe material used in loops
Water (Fluid)	0.6	Circulating fluid baseline

Step-by-Step Calculation Procedures with Worked Numerical Examples

Step 1: Determine the Building Load

The first step is to obtain the building’s peak heating/cooling load. This can be from manual calculations or software tools based on building envelope, infiltration, occupancy, and equipment. For example, a residence requires a 3.5-ton cooling capacity.

Step 2: Select Thermal Properties

From Table 1, suppose the borehole passes through moist clay soil with k_soil = 1.5 W/m·K. Assume grout k_grout = 0.8 W/m·K, PEX pipe k_pipe = 0.35 W/m·K.

Step 3: Define Geometry and Parameters

• Borehole radius (r_borehole): 0.075 m (3 inch radius)
• Grout radius (r_grout): 0.05 m
• Pipe outer radius (r_o): 0.02 m
• Pipe inner radius (r_i): 0.018 m
• Borehole length (L): to be calculated
• Fluid convective heat transfer coefficient (h_f): 1500 W/m²·K (typical high velocity water)
• Circulating fluid temperature difference (∆T): 5 K (typical design criterion)

Step 4: Calculate Individual Thermal Resistances

Fluid Convective Resistance:

A_i = 2·π·r_i·L = 2·π·0.018·L = 0.113·L m²

R_fluid = 1 / (h_f · A_i) = 1 / (1500 × 0.113·L) = 0.0059 / L K/W

Pipe Wall Resistance:

R_pipe = ln(r_o/r_i) / (2·π·k_pipe·L) = ln(0.02/0.018) / (2π × 0.35 × L) = 0.105 / (2.199 × L) = 0.0477 / L K/W

Grout Resistance:

R_grout = ln(r_grout/r_o) / (2·π·k_grout·L) = ln(0.05/0.02) / (2π × 0.8 × L) = 0.916 / (5.0265 × L) = 0.1822 / L K/W

Soil Resistance:

R_soil = ln(r_borehole/r_grout) / (2·π·k_soil·L) = ln(0.075/0.05) / (2π × 1.5 × L) = 0.405 / (9.424 × L) = 0.0430 / L K/W

Step 5: Calculate Total Thermal Resistance

R_total = (0.0059 + 0.0477 + 0.1822 + 0.0430) / L = 0.2788 / L K/W

Step 6: Calculate Required Borehole Length

Heat transfer rate q based on load: 3.5 tons × 12,000 BTU/hr/ton × 0.293 W/BTU/hr ≈ 12,300 W

For the loop, heat transfer is modeled as:

q = ∆T / R_total = ∆T × (L / 0.2788) → L = (q × 0.2788) / ∆T

Substitute values:

L = (12,300 × 0.2788) / 5 = 3436.4 m

This value is impractically long because this simplified example assumes one pipe segment. Real system designs use multiple loops, U-tubes, and vertical boreholes, often with spacing and thermal interaction included in simulation software. However, this calculation highlights the influence of textile properties and geometry on sizing.

Selection and Sizing Guidance for HVAC Applications

Proper earth loop sizing and configuration is crucial to achieve desired performance levels while optimizing installation costs. General guidelines:

• Load Matching: Use detailed HVAC load calculations (HVAC Load Calculations) accounting for peak and seasonal variations.
• Soil Testing: Conduct ASTM D5334 tests for in-situ soil thermal conductivity to improve accuracy.
• Borehole Depth and Number: Typically 150 to 400 feet depth, arranged side-by-side with spacings >15 ft to minimize thermal interference.
• Loop Configuration: Vertical boreholes for limited space, horizontal loops for large open areas; each have different excavation and installation considerations.
• Fluid Selection: Use water or antifreeze solutions (e.g., propylene glycol) depending on climate to prevent freezing and corrosion.

Table 2 displays typical loop length per ton for various ground conditions:

Table 2: Typical Loop Length per Ton depending on Soil Type

Soil Type	Thermal Conductivity (W/m·K)	Loop Length per Ton (ft)
Dry Sandy Soil	0.15 – 0.25	300 – 400 ft/ton
Moist Clay	1.2 – 1.8	150 – 250 ft/ton
Gravel	1.5 – 2.0	100 – 200 ft/ton

Best Practices and Industry Standards

Adhering to relevant standards ensures system reliability, safety, and durability:

• ASHRAE Standard 146: Guides design and installation of GSHP systems with emphasis on thermal load management and borehole thermal response testing.
• ASTM D5334: Standard Test Method for determining soil thermal conductivity using heat probe methods.
• ISO 13256-2: Specifies testing and performance for water-to-water heat pumps including geothermal applications.
• Local building codes and environmental regulations for borehole drilling and groundwater protection.

Troubleshooting and Diagnostics

Common issues affecting earth loop heat transfer include:

• Inadequate Fluid Flow: Check circulation pumps, flow meters, and valves to ensure design flow rates are met.