.Buildings.Fluid.Geothermal.Boreholes.BaseClasses.singleUTubeResistances

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.

Interface

function singleUTubeResistances
  input Modelica.Units.SI.Height hSeg "Height of the element";
  input Modelica.Units.SI.Radius rBor "Radius of the borehole";
  input Modelica.Units.SI.Radius rTub "Radius of the tube";
  input Modelica.Units.SI.Length eTub "Thickness of the tubes";
  input Modelica.Units.SI.Length xC "Shank spacing, defined as the distance between the center of a pipe and the center of the borehole";
  input Modelica.Units.SI.ThermalConductivity kSoi "Thermal conductivity of the soi";
  input Modelica.Units.SI.ThermalConductivity kFil "Thermal conductivity of the grout";
  input Modelica.Units.SI.ThermalConductivity kTub "Thermal conductivity of the tube";
  output Modelica.Units.SI.ThermalResistance Rgb "Thermal resistance between the grout zone and the borehole wall";
  output Modelica.Units.SI.ThermalResistance Rgg "Thermal resistance between the two grout zones";
  output Modelica.Units.SI.ThermalResistance RCondGro "Thermal resistance between the pipe wall ant the capacity in the grout";
  output Real x "Capacity location";
end singleUTubeResistances;

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.