functionsingleUTubeResistances

Thermal resistances for single U-tube

Information

This model computes the thermal resistances of a single-U-tube borehole using the method of Bauer et al. (2011). It also computes the fluid-to-ground thermal resistance R_b and the grout-to-grout thermal resistance R_a as defined by Hellstroem (1991) using the multipole method.

The figure below shows the thermal network set up by Bauer et al. (2011).

The different resistances are calculated as follows. The grout zone and bore hole wall thermal resistance are related as

R_gb^1U = ( 1 - x^1U ) R_g^1U.

The thermal resistance between the two grout zones are

R_gg^1U = 2 R_gb^1U ( R_ar^1U - 2 x^1U R_g^1U ) / ( 2 R_gb^1U - R_ar^1U + 2 x^1U R_g^1U ).

Thermal resistance between the pipe wall to the capacity in the grout is

R_CondGro = x^1U R_g^1U + log ( ( r_Tub + e_Tub ) /r_Tub ) / ( 2 π h_Seg k_Tub ).

The capacities are located at

x^1U =log ( (r_Bor² + 2 ( r_Tub + e_Tub ) ²)^1⁄2/ ( 2 ( r_Tub + e_Tub ) ) ) / log ( r_Bor/ ( √2 ( r_Tub + e_Tub ))).

The thermal resistance between the outer borehole wall and one tube is

R_g^1U =2 R_b ⁄ h_Seg.

The thermal resistance between the two pipe outer walls is

R_ar^1U =R_a ⁄ h_Seg.

The fluid to ground thermal resistance R_b and the grout to grout thermal resistance R_a are calculated with the multipole method (Hellstroem (1991)) as

R_b =1/ ( 4 π k_Fil ) ( log ( r_Bor/ ( r_Tub + e_Tub ) ) + log ( r_Bor/ ( 2 x_C ) ) + σ log ( r_Bor⁴/ ( r_Bor⁴ - x_C⁴ ) ) ) - 1/ ( 4 π k_Fil ) ( ( r_Tub + e_Tub ) ²/ ( 4 x_C² ) ( 1 - σ 4 x_C⁴/ ( r_Bor⁴ - x_C⁴ ) ) ² ) / ( ( 1 + β ) / ( 1 - β ) + ( r_Tub + e_Tub ) ²/ ( 4 x_C² ) ( 1 + σ 16 x_C⁴ r_Bor⁴/ ( r_Bor⁴ - x_C⁴ ) ²)).

R_a = 1/ ( π k_Fil ) ( log ( 2 x_C/r_Tub ) + σ log (( r_Bor² + x_C² ) / ( r_Bor² - x_C² ) ) ) - 1/ ( π k_Fil ) ( r_Tub²/ ( 4 x_C² ) ( 1 + σ 4 r_Bor⁴ x_C²/ ( r_Bor⁴ - x_C⁴ ) ) / ( ( 1 + β ) / ( 1 - β ) - r_Tub²/ ( 4 x_C² ) + σ 2 r_Tub² r_Bor² ( r_Bor⁴ + x_C⁴ ) / ( r_Bor⁴ - x_C⁴ ) ²)),

with σ = ( k_Fil - k_Soi ) / ( k_Fil + k_Soi ) and β = 2 π k_Fil R_CondPipe, where k_Fil and k_Soi are the conductivity of the filling material and of the ground, r_Tub+e_Tub and r_Bor are the pipe and the borehole outside radius and x_C is the shank spacing, which is equal to the distance between the center of borehole and the center of the pipe.

Note: The value of R_gg^1U may be negative as long as

1/R_gg^1U + 1/(2 R_gb^1U) > 0,

in which case the laws of thermodynamics are not violated. See Bauer et al. (2011) for details.

References

G. Hellström. Ground heat storage: thermal analyses of duct storage systems (Theory). Dept. of Mathematical Physics, University of Lund, Sweden, 1991.

D. Bauer, W. Heidemann, H. Müller-Steinhagen, and H.-J. G. Diersch. Thermal resistance and capacity models for borehole heat exchangers . International Journal Of Energy Research, 35:312–320, 2011.

Inputs

Type	Name	Description
Modelica.Units.SI.Height	hSeg	Height of the element
Modelica.Units.SI.Radius	rBor	Radius of the borehole
Modelica.Units.SI.Radius	rTub	Radius of the tube
Modelica.Units.SI.Length	eTub	Thickness of the tubes
Modelica.Units.SI.Length	xC	Shank spacing, defined as the distance between the center of a pipe and the center of the borehole
Modelica.Units.SI.ThermalConductivity	kSoi	Thermal conductivity of the soi
Modelica.Units.SI.ThermalConductivity	kFil	Thermal conductivity of the grout
Modelica.Units.SI.ThermalConductivity	kTub	Thermal conductivity of the tube

Outputs

Type	Name	Description
Modelica.Units.SI.ThermalResistance	Rgb	Thermal resistance between the grout zone and the borehole wall
Modelica.Units.SI.ThermalResistance	Rgg	Thermal resistance between the two grout zones
Modelica.Units.SI.ThermalResistance	RCondGro	Thermal resistance between the pipe wall ant the capacity in the grout
Real	x	Capacity location

Revisions

January 21, 2015 by Michael Wetter:
Fixed bug in beta being used before it was assigned.
February 14, 2014 by Michael Wetter:
Added an assert statement to test for non-physical values.
February 13, 2014 by Damien Picard:
Edit documentation and add formula for beta.
February 12, 2014, by Damien Picard:
Remove the flow dependency of the resistances, as this function calculates the conduction resistances only.
January 24, 2014, by Michael Wetter:
Revised implementation.
January 23, 2014, by Damien Picard:
First implementation.