Pipes Module¶

class pygfunction.pipes.Coaxial(pos, r_in, r_out, borehole, k_s, k_g, R_ff, R_fp, J=2)¶

Bases: SingleUTube

Class for coaxial boreholes.

Contains information regarding the physical dimensions and thermal characteristics of the pipes and the grout material, as well as methods to evaluate fluid temperatures and heat extraction rates based on the work of Hellstrom 1. Internal borehole thermal resistances are evaluated using the multipole method of Claesson and Hellstrom 2.

Notes

The expected array shapes of input parameters and outputs are documented for each class method. nInlets and nOutlets are the number of inlets and outlets to the borehole, and both are equal to 1 for a coaxial borehole. nSegments is the number of discretized segments along the borehole. nPipes is the number of pipes (i.e. the number of U-tubes) in the borehole, equal to 1. nDepths is the number of depths at which temperatures are evaluated.

The effective borehole thermal resistance is evaluated using the method of Cimmino 3. This is valid for any number of pipes.

References

1: Hellstrom, G. (1991). Ground heat storage. Thermal Analyses of Duct Storage Systems I: Theory. PhD Thesis. University of Lund, Department of Mathematical Physics. Lund, Sweden.
2: Claesson, J., & Hellstrom, G. (2011). Multipole method to calculate borehole thermal resistances in a borehole heat exchanger. HVAC&R Research, 17(6), 895-911.
3: Cimmino, M. (2019). Semi-analytical method for g-function calculation of bore fields with series- and parallel-connected boreholes. Science and Technology for the Built Environment, 25 (8), 1007-1022.

Attributes

postuple: Position (x, y) (in meters) of the pipes inside the borehole.
r_in(2,) array: Inner radii (in meters) of the coaxial pipes. The first element of the array corresponds to the inlet pipe.
r_out(2,) array: Outer radii (in meters) of the coaxial pipes. The first element of the array corresponds to the inlet pipe.
boreholeBorehole object: Borehole class object of the borehole containing the U-Tube.
k_sfloat: Soil thermal conductivity (in W/m-K).
k_gfloat: Grout thermal conductivity (in W/m-K).
R_fffloat: Fluid to fluid thermal resistance of the inner pipe to the outer pipe (in m-K/W).
R_fpfloat: Fluid to outer pipe wall thermal resistance of the outer pipe in contact with the grout (in m-K/W).
Jint, optional: Number of multipoles per pipe to evaluate the thermal resistances. Default is 2.
nPipesint: Number of U-Tubes, equals to 1.
nInletsint: Total number of pipe inlets, equals to 1.
nOutletsint: Total number of pipe outlets, equals to 1.

coefficients_borehole_heat_extraction_rate(m_flow_borehole, cp_f, nSegments, segment_ratios=None)¶

Build coefficient matrices to evaluate heat extraction rates.

Returns coefficients for the relation:

\[\mathbf{Q_b} = \mathbf{a_{in}} \mathbf{T_{f,in}} + \mathbf{a_{b}} \mathbf{T_b}\]

Parameters

m_flow_boreholefloat or (nInlets,) array: Inlet mass flow rate (in kg/s) into the borehole.
cp_ffloat or (nInlets,) array: Fluid specific isobaric heat capacity (in J/kg.degC).
nSegmentsint: Number of borehole segments.
segment_ratios(nSegments,) array, optional: Ratio of the borehole length represented by each segment. The sum of ratios must be equal to 1. If segment_ratios==None, segments of equal lengths are considered. Default is None.

Returns

a_in(nSegments, nInlets,) array: Array of coefficients for inlet fluid temperature.
a_b(nSegments, nSegments,) array: Array of coefficients for borehole wall temperatures.

coefficients_fluid_heat_extraction_rate(m_flow_borehole, cp_f, nSegments, segment_ratios=None)¶

Build coefficient matrices to evaluate heat extraction rates.

Returns coefficients for the relation:

\[\mathbf{Q_f} = \mathbf{a_{in}} \mathbf{T_{f,in}} + \mathbf{a_{b}} \mathbf{T_b}\]

Parameters

m_flow_boreholefloat or (nInlets,) array: Inlet mass flow rate (in kg/s) into the borehole.
cp_ffloat or (nInlets,) array: Fluid specific isobaric heat capacity (in J/kg.degC).
nSegmentsint: Number of borehole segments.
segment_ratios(nSegments,) array, optional: Ratio of the borehole length represented by each segment. The sum of ratios must be equal to 1. If segment_ratios==None, segments of equal lengths are considered. Default is None.

Returns

a_in(nOutlets, nInlets,) array: Array of coefficients for inlet fluid temperature.
a_b(nOutlets, nSegments,) array: Array of coefficients for borehole wall temperatures.

coefficients_inlet_temperature(m_flow_borehole, cp_f, nSegments, segment_ratios=None)¶

Build coefficient matrices to evaluate inlet fluid temperature.

Returns coefficients for the relation:

\[\mathbf{T_{f,in}} = \mathbf{a_{q,f}} \mathbf{Q_{f}} + \mathbf{a_{b}} \mathbf{T_b}\]

Parameters

m_flow_boreholefloat or (nInlets,) array: Inlet mass flow rates (in kg/s) into the borehole.
cp_ffloat or (nInlets,) array: Fluid specific isobaric heat capacity (in J/kg.degC).
nSegmentsint: Number of borehole segments.
segment_ratios(nSegments,) array, optional: Ratio of the borehole length represented by each segment. The sum of ratios must be equal to 1. If segment_ratios==None, segments of equal lengths are considered. Default is None.

Returns

a_qf(nOutlets, nInlets,) array: Array of coefficients for inlet fluid temperature.
a_b(nOutlets, nSegments,) array: Array of coefficients for borehole wall temperatures.

coefficients_outlet_temperature(m_flow_borehole, cp_f, nSegments, segment_ratios=None)¶

Build coefficient matrices to evaluate outlet fluid temperature.

Returns coefficients for the relation:

\[\mathbf{T_{f,out}} = \mathbf{a_{in}} \mathbf{T_{f,in}} + \mathbf{a_{b}} \mathbf{T_b}\]

Parameters

m_flow_boreholefloat or (nInlets,) array: Inlet mass flow rates (in kg/s) into the borehole.
cp_ffloat or (nInlets,) array: Fluid specific isobaric heat capacity (in J/kg.degC).
nSegmentsint: Number of borehole segments.
segment_ratios(nSegments,) array, optional: Ratio of the borehole length represented by each segment. The sum of ratios must be equal to 1. If segment_ratios==None, segments of equal lengths are considered. Default is None.

Returns

a_in(nOutlets, nInlets,) array: Array of coefficients for inlet fluid temperature.
a_b(nOutlets, nSegments,) array: Array of coefficients for borehole wall temperatures.

coefficients_temperature(z, m_flow_borehole, cp_f, nSegments, segment_ratios=None)¶

Build coefficient matrices to evaluate fluid temperatures at a depth (z).

Returns coefficients for the relation:

\[\mathbf{T_f}(z) = \mathbf{a_{in}} \mathbf{T_{f,in}} + \mathbf{a_{b}} \mathbf{T_b}\]

Parameters

zfloat or (nDepths,) array: Depths (in meters) to evaluate the fluid temperature coefficients.
m_flow_boreholefloat or (nInlets,) array: Inlet mass flow rate (in kg/s) into the borehole.
cp_ffloat or (nInlets,) array: Fluid specific isobaric heat capacity (in J/kg.degC).
nSegmentsint: Number of borehole segments.
segment_ratios(nSegments,) array, optional: Ratio of the borehole length represented by each segment. The sum of ratios must be equal to 1. If segment_ratios==None, segments of equal lengths are considered. Default is None.

Returns

a_in
(2*nPipes, nInlets,) array, or (nDepths, 2*nPipes, nInlets,) array: Array of coefficients for inlet fluid temperature.
a_b
(2*nPipes, nSegments,) array, or (nDepths, 2*nPipes, nSegments,) array: Array of coefficients for borehole wall temperatures.

effective_borehole_thermal_resistance(m_flow_borehole, cp_f)¶

Evaluate the effective borehole thermal resistance, defined by:

\[ \begin{align}\begin{aligned}\frac{Q_b}{H} = \frac{T^*_b - \bar{T}_f}{R^*_b}\\\bar{T}_f = \frac{1}{2}(T_{f,in} + T_{f,out})\end{aligned}\end{align} \]

where \(Q_b\) is the borehole heat extraction rate (in Watts), \(H\) is the borehole length, \(T^*_b\) is the effective borehole wall temperature, \(R^*_b\) is the effective borehole thermal resistance, \(T_{f,in}\) is the inlet fluid temperature, and \(T_{f,out}\) is the outlet fluid temperature.

Parameters

m_flow_boreholefloat: Fluid mass flow rate (in kg/s) into the borehole.
cp_ffloat: Fluid specific isobaric heat capacity (in J/kg.K)

Returns

R_bfloat: Effective borehole thermal resistance (in m.K/W).

get_borehole_heat_extraction_rate(T_f_in, T_b, m_flow_borehole, cp_f, segment_ratios=None)¶

Returns the heat extraction rates of the borehole.

Parameters

T_f_infloat or (nInlets,) array: Inlet fluid temperatures (in Celsius).
T_bfloat or (nSegments,) array: Borehole wall temperatures (in Celsius).
m_flow_boreholefloat or (nInlets,) array: Inlet mass flow rates (in kg/s) into the borehole.
cp_ffloat or (nInlets,) array: Fluid specific isobaric heat capacity (in J/kg.degC).
segment_ratios(nSegments,) array, optional: Ratio of the borehole length represented by each segment. The sum of ratios must be equal to 1. If segment_ratios==None, segments of equal lengths are considered. Default is None.

Returns

Q_bfloat or (nSegments,) array: Heat extraction rates along each borehole segment (in Watts). The returned type corresponds to the type of the parameter T_b.

get_fluid_heat_extraction_rate(T_f_in, T_b, m_flow_borehole, cp_f, segment_ratios=None)¶

Returns the heat extraction rates of the borehole.

Parameters

T_f_infloat or (nInlets,) array: Inlet fluid temperatures (in Celsius).
T_bfloat or (nSegments,) array: Borehole wall temperatures (in Celsius).
m_flow_boreholefloat or (nInlets,) array: Inlet mass flow rates (in kg/s) into the borehole.
cp_ffloat or (nInlets,) array: Fluid specific isobaric heat capacity (in J/kg.degC).
segment_ratios(nSegments,) array, optional: Ratio of the borehole length represented by each segment. The sum of ratios must be equal to 1. If segment_ratios==None, segments of equal lengths are considered. Default is None.

Returns

Q_ffloat or (nOutlets,) array: Heat extraction rates from each fluid circuit (in Watts). The returned type corresponds to the type of the parameter T_f_in.

get_inlet_temperature(Q_f, T_b, m_flow_borehole, cp_f, segment_ratios=None)¶

Returns the inlet fluid temperatures of the borehole.

Parameters

Q_ffloat or (nInlets,) array: Heat extraction from the fluid circuits (in Watts).
T_bfloat or (nSegments,) array: Borehole wall temperatures (in Celsius).
m_flow_boreholefloat or (nInlets,) array: Inlet mass flow rates (in kg/s) into the borehole.
cp_ffloat or (nInlets,) array: Fluid specific isobaric heat capacity (in J/kg.degC).
segment_ratios(nSegments,) array, optional: Ratio of the borehole length represented by each segment. The sum of ratios must be equal to 1. If segment_ratios==None, segments of equal lengths are considered. Default is None.

Returns

T_infloat or (nOutlets,) array: Inlet fluid temperatures (in Celsius) into each inlet pipe. The returned type corresponds to the type of the parameter Q_f.

get_outlet_temperature(T_f_in, T_b, m_flow_borehole, cp_f, segment_ratios=None)¶

Returns the outlet fluid temperatures of the borehole.

Parameters

T_f_infloat or (nInlets,) array: Inlet fluid temperatures (in Celsius).
T_bfloat or (nSegments,) array: Borehole wall temperatures (in Celsius).
m_flow_boreholefloat or (nInlets,) array: Inlet mass flow rates (in kg/s) into the borehole.
cp_ffloat or (nInlets,) array: Fluid specific isobaric heat capacity (in J/kg.degC).
segment_ratios(nSegments,) array, optional: Ratio of the borehole length represented by each segment. The sum of ratios must be equal to 1. If segment_ratios==None, segments of equal lengths are considered. Default is None.

Returns

T_f_outfloat or (nOutlets,) array: Outlet fluid temperatures (in Celsius) from each outlet pipe. The returned type corresponds to the type of the parameter T_f_in.

get_temperature(z, T_f_in, T_b, m_flow_borehole, cp_f, segment_ratios=None)¶

Returns the fluid temperatures of the borehole at a depth (z).

Parameters

zfloat or (nDepths,) array: Depths (in meters) to evaluate the fluid temperatures.
T_f_infloat or (nInlets,) array: Inlet fluid temperatures (in Celsius).
T_bfloat or (nSegments,) array: Borehole wall temperatures (in Celsius).
m_flow_boreholefloat or (nInlets,) array: Inlet mass flow rates (in kg/s) into the borehole.
cp_ffloat or (nInlets,) array: Fluid specific isobaric heat capacity (in J/kg.degC).
segment_ratios(nSegments,) array, optional: Ratio of the borehole length represented by each segment. The sum of ratios must be equal to 1. If segment_ratios==None, segments of equal lengths are considered. Default is None.

Returns

T_f(2*nPipes,) or (nDepths, 2*nPipes,) array: Fluid temperature (in Celsius) in each pipe. The returned shape depends on the type of the parameter z.

get_total_heat_extraction_rate(T_f_in, T_b, m_flow_borehole, cp_f, segment_ratios=None)¶

Returns the total heat extraction rate of the borehole.

Parameters

T_f_infloat or (nInlets,) array: Inlet fluid temperatures (in Celsius).
T_bfloat or (nSegments,) array: Borehole wall temperatures (in Celsius).
m_flow_boreholefloat or (nInlets,) array: Inlet mass flow rates (in kg/s) into the borehole.
cp_ffloat or (nInlets,) array: Fluid specific isobaric heat capacity (in J/kg.degC).
segment_ratios(nSegments,) array, optional: Ratio of the borehole length represented by each segment. The sum of ratios must be equal to 1. If segment_ratios==None, segments of equal lengths are considered. Default is None.

Returns

Q_tfloat: Total net heat extraction rate of the borehole (in Watts).

local_borehole_thermal_resistance()¶

Evaluate the local (cross-sectional) borehole thermal resistance, defined by:

\[Q'_b(z) = \frac{T_b(z) - \bar{T}_f(z)}{R_b}\]

where \(Q'_b(z)\) is the borehole heat extraction rate per unit depth at a depth \((z)\), \(T_b(z)\) is the borehole wall temperature, \(\bar{T}_f(z)\) is the arithmetic mean fluid temperature and \(R_b\) is the local borehole thermal resistance.

Returns

R_bfloat: Local borehole thermal resistance (in m.K/W).

update_thermal_resistances(R_ff, R_fp)¶

Update the delta-circuit of thermal resistances.

This methods updates the values of the delta-circuit thermal resistances based on the provided fluid to fluid and fluid to outer pipe wall thermal resistances.

Parameters

R_fffloat: Fluid to fluid thermal resistance of the inner pipe to the outer pipe (in m-K/W).
R_fpfloat: Fluid to outer pipe wall thermal resistance of the outer pipe in contact with the grout (in m-K/W).

visualize_pipes()¶

Plot the cross-section view of the borehole.

Returns

figfigure: Figure object (matplotlib).

class pygfunction.pipes.IndependentMultipleUTube(pos, r_in, r_out, borehole, k_s, k_g, R_fp, nPipes, J=2)¶

Bases: MultipleUTube

Class for multiple U-Tube boreholes with independent U-tubes.

Contains information regarding the physical dimensions and thermal characteristics of the pipes and the grout material, as well as methods to evaluate fluid temperatures and heat extraction rates based on the work of Cimmino 4 for boreholes with any number of U-tubes. Internal borehole thermal resistances are evaluated using the multipole method of Claesson and Hellstrom 5.

Notes

The expected array shapes of input parameters and outputs are documented for each class method. nInlets and nOutlets are the number of inlets and outlets to the borehole, and both are equal to the number of pipes. nSegments is the number of discretized segments along the borehole. nPipes is the number of pipes (i.e. the number of U-tubes) in the borehole. nDepths is the number of depths at which temperatures are evaluated.

References

4: Cimmino, M. (2016). Fluid and borehole wall temperature profiles in vertical geothermal boreholes with multiple U-tubes. Renewable Energy, 96, 137-147.
5: Claesson, J., & Hellstrom, G. (2011). Multipole method to calculate borehole thermal resistances in a borehole heat exchanger. HVAC&R Research, 17(6), 895-911.

Attributes

poslist of tuples: Position (x, y) (in meters) of the pipes inside the borehole.
r_infloat: Inner radius (in meters) of the U-Tube pipes.
r_outfloat: Outer radius (in meters) of the U-Tube pipes.
boreholeBorehole object: Borehole class object of the borehole containing the U-Tube.
k_sfloat: Soil thermal conductivity (in W/m-K).
k_gfloat: Grout thermal conductivity (in W/m-K).
R_fpfloat: Fluid to outer pipe wall thermal resistance (m-K/W).
Jint, optional: Number of multipoles per pipe to evaluate the thermal resistances. Default is 2.
nPipesint: Number of U-Tubes.
nInletsint: Total number of pipe inlets, equals to nPipes.
nOutletsint: Total number of pipe outlets, equals to nPipes.

coefficients_borehole_heat_extraction_rate(m_flow_borehole, cp_f, nSegments, segment_ratios=None)¶

Build coefficient matrices to evaluate heat extraction rates.

Returns coefficients for the relation:

\[\mathbf{Q_b} = \mathbf{a_{in}} \mathbf{T_{f,in}} + \mathbf{a_{b}} \mathbf{T_b}\]

Parameters

m_flow_boreholefloat or (nInlets,) array: Inlet mass flow rate (in kg/s) into the borehole.
cp_ffloat or (nInlets,) array: Fluid specific isobaric heat capacity (in J/kg.degC).
nSegmentsint: Number of borehole segments.
segment_ratios(nSegments,) array, optional: Ratio of the borehole length represented by each segment. The sum of ratios must be equal to 1. If segment_ratios==None, segments of equal lengths are considered. Default is None.

Returns

a_in(nSegments, nInlets,) array: Array of coefficients for inlet fluid temperature.
a_b(nSegments, nSegments,) array: Array of coefficients for borehole wall temperatures.

coefficients_fluid_heat_extraction_rate(m_flow_borehole, cp_f, nSegments, segment_ratios=None)¶

Build coefficient matrices to evaluate heat extraction rates.

Returns coefficients for the relation:

\[\mathbf{Q_f} = \mathbf{a_{in}} \mathbf{T_{f,in}} + \mathbf{a_{b}} \mathbf{T_b}\]

Parameters

m_flow_boreholefloat or (nInlets,) array: Inlet mass flow rate (in kg/s) into the borehole.
cp_ffloat or (nInlets,) array: Fluid specific isobaric heat capacity (in J/kg.degC).
nSegmentsint: Number of borehole segments.
segment_ratios(nSegments,) array, optional: Ratio of the borehole length represented by each segment. The sum of ratios must be equal to 1. If segment_ratios==None, segments of equal lengths are considered. Default is None.

Returns

a_in(nOutlets, nInlets,) array: Array of coefficients for inlet fluid temperature.
a_b(nOutlets, nSegments,) array: Array of coefficients for borehole wall temperatures.

coefficients_inlet_temperature(m_flow_borehole, cp_f, nSegments, segment_ratios=None)¶

Build coefficient matrices to evaluate inlet fluid temperature.

Returns coefficients for the relation:

\[\mathbf{T_{f,in}} = \mathbf{a_{q,f}} \mathbf{Q_{f}} + \mathbf{a_{b}} \mathbf{T_b}\]

Parameters

m_flow_boreholefloat or (nInlets,) array: Inlet mass flow rates (in kg/s) into the borehole.
cp_ffloat or (nInlets,) array: Fluid specific isobaric heat capacity (in J/kg.degC).
nSegmentsint: Number of borehole segments.
segment_ratios(nSegments,) array, optional: Ratio of the borehole length represented by each segment. The sum of ratios must be equal to 1. If segment_ratios==None, segments of equal lengths are considered. Default is None.

Returns

a_qf(nOutlets, nInlets,) array: Array of coefficients for inlet fluid temperature.
a_b(nOutlets, nSegments,) array: Array of coefficients for borehole wall temperatures.

coefficients_outlet_temperature(m_flow_borehole, cp_f, nSegments, segment_ratios=None)¶

Build coefficient matrices to evaluate outlet fluid temperature.

Returns coefficients for the relation:

\[\mathbf{T_{f,out}} = \mathbf{a_{in}} \mathbf{T_{f,in}} + \mathbf{a_{b}} \mathbf{T_b}\]

Parameters

m_flow_boreholefloat or (nInlets,) array: Inlet mass flow rates (in kg/s) into the borehole.
cp_ffloat or (nInlets,) array: Fluid specific isobaric heat capacity (in J/kg.degC).
nSegmentsint: Number of borehole segments.
segment_ratios(nSegments,) array, optional: Ratio of the borehole length represented by each segment. The sum of ratios must be equal to 1. If segment_ratios==None, segments of equal lengths are considered. Default is None.

Returns

a_in(nOutlets, nInlets,) array: Array of coefficients for inlet fluid temperature.
a_b(nOutlets, nSegments,) array: Array of coefficients for borehole wall temperatures.

coefficients_temperature(z, m_flow_borehole, cp_f, nSegments, segment_ratios=None)¶

Build coefficient matrices to evaluate fluid temperatures at a depth (z).

Returns coefficients for the relation:

\[\mathbf{T_f}(z) = \mathbf{a_{in}} \mathbf{T_{f,in}} + \mathbf{a_{b}} \mathbf{T_b}\]

Parameters

zfloat or (nDepths,) array: Depths (in meters) to evaluate the fluid temperature coefficients.
m_flow_boreholefloat or (nInlets,) array: Inlet mass flow rate (in kg/s) into the borehole.
cp_ffloat or (nInlets,) array: Fluid specific isobaric heat capacity (in J/kg.degC).
nSegmentsint: Number of borehole segments.
segment_ratios(nSegments,) array, optional: Ratio of the borehole length represented by each segment. The sum of ratios must be equal to 1. If segment_ratios==None, segments of equal lengths are considered. Default is None.

Returns

a_in
(2*nPipes, nInlets,) array, or (nDepths, 2*nPipes, nInlets,) array: Array of coefficients for inlet fluid temperature.
a_b
(2*nPipes, nSegments,) array, or (nDepths, 2*nPipes, nSegments,) array: Array of coefficients for borehole wall temperatures.

effective_borehole_thermal_resistance(m_flow_borehole, cp_f)¶

Evaluate the effective borehole thermal resistance, defined by:

\[ \begin{align}\begin{aligned}\frac{Q_b}{H} = \frac{T^*_b - \bar{T}_f}{R^*_b}\\\bar{T}_f = \frac{1}{2}(T_{f,in} + T_{f,out})\end{aligned}\end{align} \]

where \(Q_b\) is the borehole heat extraction rate (in Watts), \(H\) is the borehole length, \(T^*_b\) is the effective borehole wall temperature, \(R^*_b\) is the effective borehole thermal resistance, \(T_{f,in}\) is the inlet fluid temperature, and \(T_{f,out}\) is the outlet fluid temperature.

Parameters

m_flow_boreholefloat: Fluid mass flow rate (in kg/s) into the borehole.
cp_ffloat: Fluid specific isobaric heat capacity (in J/kg.K)

Returns

R_bfloat: Effective borehole thermal resistance (in m.K/W).

get_borehole_heat_extraction_rate(T_f_in, T_b, m_flow_borehole, cp_f, segment_ratios=None)¶

Returns the heat extraction rates of the borehole.

Parameters

T_f_infloat or (nInlets,) array: Inlet fluid temperatures (in Celsius).
T_bfloat or (nSegments,) array: Borehole wall temperatures (in Celsius).
m_flow_boreholefloat or (nInlets,) array: Inlet mass flow rates (in kg/s) into the borehole.
cp_ffloat or (nInlets,) array: Fluid specific isobaric heat capacity (in J/kg.degC).
segment_ratios(nSegments,) array, optional: Ratio of the borehole length represented by each segment. The sum of ratios must be equal to 1. If segment_ratios==None, segments of equal lengths are considered. Default is None.

Returns

Q_bfloat or (nSegments,) array: Heat extraction rates along each borehole segment (in Watts). The returned type corresponds to the type of the parameter T_b.

get_fluid_heat_extraction_rate(T_f_in, T_b, m_flow_borehole, cp_f, segment_ratios=None)¶

Returns the heat extraction rates of the borehole.

Parameters

T_f_infloat or (nInlets,) array: Inlet fluid temperatures (in Celsius).
T_bfloat or (nSegments,) array: Borehole wall temperatures (in Celsius).
m_flow_boreholefloat or (nInlets,) array: Inlet mass flow rates (in kg/s) into the borehole.
cp_ffloat or (nInlets,) array: Fluid specific isobaric heat capacity (in J/kg.degC).
segment_ratios(nSegments,) array, optional: Ratio of the borehole length represented by each segment. The sum of ratios must be equal to 1. If segment_ratios==None, segments of equal lengths are considered. Default is None.

Returns

Q_ffloat or (nOutlets,) array: Heat extraction rates from each fluid circuit (in Watts). The returned type corresponds to the type of the parameter T_f_in.

get_inlet_temperature(Q_f, T_b, m_flow_borehole, cp_f, segment_ratios=None)¶

Returns the inlet fluid temperatures of the borehole.

Parameters

Q_ffloat or (nInlets,) array: Heat extraction from the fluid circuits (in Watts).
T_bfloat or (nSegments,) array: Borehole wall temperatures (in Celsius).
m_flow_boreholefloat or (nInlets,) array: Inlet mass flow rates (in kg/s) into the borehole.
cp_ffloat or (nInlets,) array: Fluid specific isobaric heat capacity (in J/kg.degC).
segment_ratios(nSegments,) array, optional: Ratio of the borehole length represented by each segment. The sum of ratios must be equal to 1. If segment_ratios==None, segments of equal lengths are considered. Default is None.

Returns

T_infloat or (nOutlets,) array: Inlet fluid temperatures (in Celsius) into each inlet pipe. The returned type corresponds to the type of the parameter Q_f.

get_outlet_temperature(T_f_in, T_b, m_flow_borehole, cp_f, segment_ratios=None)¶

Returns the outlet fluid temperatures of the borehole.

Parameters

T_f_infloat or (nInlets,) array: Inlet fluid temperatures (in Celsius).
T_bfloat or (nSegments,) array: Borehole wall temperatures (in Celsius).
m_flow_boreholefloat or (nInlets,) array: Inlet mass flow rates (in kg/s) into the borehole.
cp_ffloat or (nInlets,) array: Fluid specific isobaric heat capacity (in J/kg.degC).
segment_ratios(nSegments,) array, optional: Ratio of the borehole length represented by each segment. The sum of ratios must be equal to 1. If segment_ratios==None, segments of equal lengths are considered. Default is None.

Returns

T_f_outfloat or (nOutlets,) array: Outlet fluid temperatures (in Celsius) from each outlet pipe. The returned type corresponds to the type of the parameter T_f_in.

get_temperature(z, T_f_in, T_b, m_flow_borehole, cp_f, segment_ratios=None)¶

Returns the fluid temperatures of the borehole at a depth (z).

Parameters

zfloat or (nDepths,) array: Depths (in meters) to evaluate the fluid temperatures.
T_f_infloat or (nInlets,) array: Inlet fluid temperatures (in Celsius).
T_bfloat or (nSegments,) array: Borehole wall temperatures (in Celsius).
m_flow_boreholefloat or (nInlets,) array: Inlet mass flow rates (in kg/s) into the borehole.
cp_ffloat or (nInlets,) array: Fluid specific isobaric heat capacity (in J/kg.degC).
segment_ratios(nSegments,) array, optional: Ratio of the borehole length represented by each segment. The sum of ratios must be equal to 1. If segment_ratios==None, segments of equal lengths are considered. Default is None.

Returns

T_f(2*nPipes,) or (nDepths, 2*nPipes,) array: Fluid temperature (in Celsius) in each pipe. The returned shape depends on the type of the parameter z.

get_total_heat_extraction_rate(T_f_in, T_b, m_flow_borehole, cp_f, segment_ratios=None)¶

Returns the total heat extraction rate of the borehole.

Parameters

T_f_infloat or (nInlets,) array: Inlet fluid temperatures (in Celsius).
T_bfloat or (nSegments,) array: Borehole wall temperatures (in Celsius).
m_flow_boreholefloat or (nInlets,) array: Inlet mass flow rates (in kg/s) into the borehole.
cp_ffloat or (nInlets,) array: Fluid specific isobaric heat capacity (in J/kg.degC).
segment_ratios(nSegments,) array, optional: Ratio of the borehole length represented by each segment. The sum of ratios must be equal to 1. If segment_ratios==None, segments of equal lengths are considered. Default is None.

Returns

Q_tfloat: Total net heat extraction rate of the borehole (in Watts).

local_borehole_thermal_resistance()¶

Evaluate the local (cross-sectional) borehole thermal resistance, defined by:

\[Q'_b(z) = \frac{T_b(z) - \bar{T}_f(z)}{R_b}\]

where \(Q'_b(z)\) is the borehole heat extraction rate per unit depth at a depth \((z)\), \(T_b(z)\) is the borehole wall temperature, \(\bar{T}_f(z)\) is the arithmetic mean fluid temperature and \(R_b\) is the local borehole thermal resistance.

Returns

R_bfloat: Local borehole thermal resistance (in m.K/W).

update_thermal_resistances(R_fp)¶

Update the delta-circuit of thermal resistances.

This methods updates the values of the delta-circuit thermal resistances based on the provided fluid to outer pipe wall thermal resistance.

Parameters

R_fpfloat: Fluid to outer pipe wall thermal resistance (m-K/W).

visualize_pipes()¶

Plot the cross-section view of the borehole.

Returns

figfigure: Figure object (matplotlib).

class pygfunction.pipes.MultipleUTube(pos, r_in, r_out, borehole, k_s, k_g, R_fp, nPipes, config='parallel', J=2)¶

Bases: _BasePipe

Class for multiple U-Tube boreholes.

Contains information regarding the physical dimensions and thermal characteristics of the pipes and the grout material, as well as methods to evaluate fluid temperatures and heat extraction rates based on the work of Cimmino 6 for boreholes with any number of U-tubes. Internal borehole thermal resistances are evaluated using the multipole method of Claesson and Hellstrom 7.

Notes

The expected array shapes of input parameters and outputs are documented for each class method. nInlets and nOutlets are the number of inlets and outlets to the borehole, and both are equal to 1 for a multiple U-tube borehole. nSegments is the number of discretized segments along the borehole. nPipes is the number of pipes (i.e. the number of U-tubes) in the borehole. nDepths is the number of depths at which temperatures are evaluated.

The effective borehole thermal resistance is evaluated using the method of Cimmino 8. This is valid for any number of pipes.

References

6: Cimmino, M. (2016). Fluid and borehole wall temperature profiles in vertical geothermal boreholes with multiple U-tubes. Renewable Energy, 96, 137-147.
7: Claesson, J., & Hellstrom, G. (2011). Multipole method to calculate borehole thermal resistances in a borehole heat exchanger. HVAC&R Research, 17(6), 895-911.
8: Cimmino, M. (2019). Semi-analytical method for g-function calculation of bore fields with series- and parallel-connected boreholes. Science and Technology for the Built Environment, 25 (8), 1007-1022.

Attributes

poslist of tuples

Position (x, y) (in meters) of the pipes inside the borehole.

r_infloat

Inner radius (in meters) of the U-Tube pipes.

r_outfloat

Outer radius (in meters) of the U-Tube pipes.

boreholeBorehole object

Borehole class object of the borehole containing the U-Tube.

k_sfloat

Soil thermal conductivity (in W/m-K).

k_gfloat

Grout thermal conductivity (in W/m-K).

R_fpfloat

Fluid to outer pipe wall thermal resistance (m-K/W).

Jint, optional

Number of multipoles per pipe to evaluate the thermal resistances. Default is 2.

nPipesint

Number of U-Tubes.

configstr, defaults to ‘parallel’

Configuration of the U-Tube pipes:: ‘parallel’ : U-tubes are connected in parallel. ‘series’ : U-tubes are connected in series.

nInletsint

Total number of pipe inlets, equals to 1.

nOutletsint

Total number of pipe outlets, equals to 1.

coefficients_borehole_heat_extraction_rate(m_flow_borehole, cp_f, nSegments, segment_ratios=None)¶

Build coefficient matrices to evaluate heat extraction rates.

Returns coefficients for the relation:

\[\mathbf{Q_b} = \mathbf{a_{in}} \mathbf{T_{f,in}} + \mathbf{a_{b}} \mathbf{T_b}\]

Parameters

m_flow_boreholefloat or (nInlets,) array: Inlet mass flow rate (in kg/s) into the borehole.
cp_ffloat or (nInlets,) array: Fluid specific isobaric heat capacity (in J/kg.degC).
nSegmentsint: Number of borehole segments.
segment_ratios(nSegments,) array, optional: Ratio of the borehole length represented by each segment. The sum of ratios must be equal to 1. If segment_ratios==None, segments of equal lengths are considered. Default is None.

Returns

a_in(nSegments, nInlets,) array: Array of coefficients for inlet fluid temperature.
a_b(nSegments, nSegments,) array: Array of coefficients for borehole wall temperatures.

coefficients_fluid_heat_extraction_rate(m_flow_borehole, cp_f, nSegments, segment_ratios=None)¶

Build coefficient matrices to evaluate heat extraction rates.

Returns coefficients for the relation:

\[\mathbf{Q_f} = \mathbf{a_{in}} \mathbf{T_{f,in}} + \mathbf{a_{b}} \mathbf{T_b}\]

Parameters

m_flow_boreholefloat or (nInlets,) array: Inlet mass flow rate (in kg/s) into the borehole.
cp_ffloat or (nInlets,) array: Fluid specific isobaric heat capacity (in J/kg.degC).
nSegmentsint: Number of borehole segments.
segment_ratios(nSegments,) array, optional: Ratio of the borehole length represented by each segment. The sum of ratios must be equal to 1. If segment_ratios==None, segments of equal lengths are considered. Default is None.

Returns

a_in(nOutlets, nInlets,) array: Array of coefficients for inlet fluid temperature.
a_b(nOutlets, nSegments,) array: Array of coefficients for borehole wall temperatures.

coefficients_inlet_temperature(m_flow_borehole, cp_f, nSegments, segment_ratios=None)¶

Build coefficient matrices to evaluate inlet fluid temperature.

Returns coefficients for the relation:

\[\mathbf{T_{f,in}} = \mathbf{a_{q,f}} \mathbf{Q_{f}} + \mathbf{a_{b}} \mathbf{T_b}\]

Parameters

m_flow_boreholefloat or (nInlets,) array: Inlet mass flow rates (in kg/s) into the borehole.
cp_ffloat or (nInlets,) array: Fluid specific isobaric heat capacity (in J/kg.degC).
nSegmentsint: Number of borehole segments.
segment_ratios(nSegments,) array, optional: Ratio of the borehole length represented by each segment. The sum of ratios must be equal to 1. If segment_ratios==None, segments of equal lengths are considered. Default is None.

Returns

a_qf(nOutlets, nInlets,) array: Array of coefficients for inlet fluid temperature.
a_b(nOutlets, nSegments,) array: Array of coefficients for borehole wall temperatures.

coefficients_outlet_temperature(m_flow_borehole, cp_f, nSegments, segment_ratios=None)¶

Build coefficient matrices to evaluate outlet fluid temperature.

Returns coefficients for the relation:

\[\mathbf{T_{f,out}} = \mathbf{a_{in}} \mathbf{T_{f,in}} + \mathbf{a_{b}} \mathbf{T_b}\]

Parameters

m_flow_boreholefloat or (nInlets,) array: Inlet mass flow rates (in kg/s) into the borehole.
cp_ffloat or (nInlets,) array: Fluid specific isobaric heat capacity (in J/kg.degC).
nSegmentsint: Number of borehole segments.
segment_ratios(nSegments,) array, optional: Ratio of the borehole length represented by each segment. The sum of ratios must be equal to 1. If segment_ratios==None, segments of equal lengths are considered. Default is None.

Returns

a_in(nOutlets, nInlets,) array: Array of coefficients for inlet fluid temperature.
a_b(nOutlets, nSegments,) array: Array of coefficients for borehole wall temperatures.

coefficients_temperature(z, m_flow_borehole, cp_f, nSegments, segment_ratios=None)¶

Build coefficient matrices to evaluate fluid temperatures at a depth (z).

Returns coefficients for the relation:

\[\mathbf{T_f}(z) = \mathbf{a_{in}} \mathbf{T_{f,in}} + \mathbf{a_{b}} \mathbf{T_b}\]

Parameters

zfloat or (nDepths,) array: Depths (in meters) to evaluate the fluid temperature coefficients.
m_flow_boreholefloat or (nInlets,) array: Inlet mass flow rate (in kg/s) into the borehole.
cp_ffloat or (nInlets,) array: Fluid specific isobaric heat capacity (in J/kg.degC).
nSegmentsint: Number of borehole segments.
segment_ratios(nSegments,) array, optional: Ratio of the borehole length represented by each segment. The sum of ratios must be equal to 1. If segment_ratios==None, segments of equal lengths are considered. Default is None.

Returns

a_in
(2*nPipes, nInlets,) array, or (nDepths, 2*nPipes, nInlets,) array: Array of coefficients for inlet fluid temperature.
a_b
(2*nPipes, nSegments,) array, or (nDepths, 2*nPipes, nSegments,) array: Array of coefficients for borehole wall temperatures.

effective_borehole_thermal_resistance(m_flow_borehole, cp_f)¶

Evaluate the effective borehole thermal resistance, defined by:

\[ \begin{align}\begin{aligned}\frac{Q_b}{H} = \frac{T^*_b - \bar{T}_f}{R^*_b}\\\bar{T}_f = \frac{1}{2}(T_{f,in} + T_{f,out})\end{aligned}\end{align} \]

where \(Q_b\) is the borehole heat extraction rate (in Watts), \(H\) is the borehole length, \(T^*_b\) is the effective borehole wall temperature, \(R^*_b\) is the effective borehole thermal resistance, \(T_{f,in}\) is the inlet fluid temperature, and \(T_{f,out}\) is the outlet fluid temperature.

Parameters

m_flow_boreholefloat: Fluid mass flow rate (in kg/s) into the borehole.
cp_ffloat: Fluid specific isobaric heat capacity (in J/kg.K)

Returns

R_bfloat: Effective borehole thermal resistance (in m.K/W).

get_borehole_heat_extraction_rate(T_f_in, T_b, m_flow_borehole, cp_f, segment_ratios=None)¶

Returns the heat extraction rates of the borehole.

Parameters

T_f_infloat or (nInlets,) array: Inlet fluid temperatures (in Celsius).
T_bfloat or (nSegments,) array: Borehole wall temperatures (in Celsius).
m_flow_boreholefloat or (nInlets,) array: Inlet mass flow rates (in kg/s) into the borehole.
cp_ffloat or (nInlets,) array: Fluid specific isobaric heat capacity (in J/kg.degC).
segment_ratios(nSegments,) array, optional: Ratio of the borehole length represented by each segment. The sum of ratios must be equal to 1. If segment_ratios==None, segments of equal lengths are considered. Default is None.

Returns

Q_bfloat or (nSegments,) array: Heat extraction rates along each borehole segment (in Watts). The returned type corresponds to the type of the parameter T_b.

get_fluid_heat_extraction_rate(T_f_in, T_b, m_flow_borehole, cp_f, segment_ratios=None)¶

Returns the heat extraction rates of the borehole.

Parameters

T_f_infloat or (nInlets,) array: Inlet fluid temperatures (in Celsius).
T_bfloat or (nSegments,) array: Borehole wall temperatures (in Celsius).
m_flow_boreholefloat or (nInlets,) array: Inlet mass flow rates (in kg/s) into the borehole.
cp_ffloat or (nInlets,) array: Fluid specific isobaric heat capacity (in J/kg.degC).
segment_ratios(nSegments,) array, optional: Ratio of the borehole length represented by each segment. The sum of ratios must be equal to 1. If segment_ratios==None, segments of equal lengths are considered. Default is None.

Returns

Q_ffloat or (nOutlets,) array: Heat extraction rates from each fluid circuit (in Watts). The returned type corresponds to the type of the parameter T_f_in.

get_inlet_temperature(Q_f, T_b, m_flow_borehole, cp_f, segment_ratios=None)¶

Returns the inlet fluid temperatures of the borehole.

Parameters

Q_ffloat or (nInlets,) array: Heat extraction from the fluid circuits (in Watts).
T_bfloat or (nSegments,) array: Borehole wall temperatures (in Celsius).
m_flow_boreholefloat or (nInlets,) array: Inlet mass flow rates (in kg/s) into the borehole.
cp_ffloat or (nInlets,) array: Fluid specific isobaric heat capacity (in J/kg.degC).
segment_ratios(nSegments,) array, optional: Ratio of the borehole length represented by each segment. The sum of ratios must be equal to 1. If segment_ratios==None, segments of equal lengths are considered. Default is None.

Returns

T_infloat or (nOutlets,) array: Inlet fluid temperatures (in Celsius) into each inlet pipe. The returned type corresponds to the type of the parameter Q_f.

get_outlet_temperature(T_f_in, T_b, m_flow_borehole, cp_f, segment_ratios=None)¶

Returns the outlet fluid temperatures of the borehole.

Parameters

T_f_infloat or (nInlets,) array: Inlet fluid temperatures (in Celsius).
T_bfloat or (nSegments,) array: Borehole wall temperatures (in Celsius).
m_flow_boreholefloat or (nInlets,) array: Inlet mass flow rates (in kg/s) into the borehole.
cp_ffloat or (nInlets,) array: Fluid specific isobaric heat capacity (in J/kg.degC).
segment_ratios(nSegments,) array, optional: Ratio of the borehole length represented by each segment. The sum of ratios must be equal to 1. If segment_ratios==None, segments of equal lengths are considered. Default is None.

Returns

T_f_outfloat or (nOutlets,) array: Outlet fluid temperatures (in Celsius) from each outlet pipe. The returned type corresponds to the type of the parameter T_f_in.

get_temperature(z, T_f_in, T_b, m_flow_borehole, cp_f, segment_ratios=None)¶

Returns the fluid temperatures of the borehole at a depth (z).

Parameters

zfloat or (nDepths,) array: Depths (in meters) to evaluate the fluid temperatures.
T_f_infloat or (nInlets,) array: Inlet fluid temperatures (in Celsius).
T_bfloat or (nSegments,) array: Borehole wall temperatures (in Celsius).
m_flow_boreholefloat or (nInlets,) array: Inlet mass flow rates (in kg/s) into the borehole.
cp_ffloat or (nInlets,) array: Fluid specific isobaric heat capacity (in J/kg.degC).
segment_ratios(nSegments,) array, optional: Ratio of the borehole length represented by each segment. The sum of ratios must be equal to 1. If segment_ratios==None, segments of equal lengths are considered. Default is None.

Returns

T_f(2*nPipes,) or (nDepths, 2*nPipes,) array: Fluid temperature (in Celsius) in each pipe. The returned shape depends on the type of the parameter z.

get_total_heat_extraction_rate(T_f_in, T_b, m_flow_borehole, cp_f, segment_ratios=None)¶

Returns the total heat extraction rate of the borehole.

Parameters

T_f_infloat or (nInlets,) array: Inlet fluid temperatures (in Celsius).
T_bfloat or (nSegments,) array: Borehole wall temperatures (in Celsius).
m_flow_boreholefloat or (nInlets,) array: Inlet mass flow rates (in kg/s) into the borehole.
cp_ffloat or (nInlets,) array: Fluid specific isobaric heat capacity (in J/kg.degC).
segment_ratios(nSegments,) array, optional: Ratio of the borehole length represented by each segment. The sum of ratios must be equal to 1. If segment_ratios==None, segments of equal lengths are considered. Default is None.

Returns

Q_tfloat: Total net heat extraction rate of the borehole (in Watts).

local_borehole_thermal_resistance()¶

Evaluate the local (cross-sectional) borehole thermal resistance, defined by:

\[Q'_b(z) = \frac{T_b(z) - \bar{T}_f(z)}{R_b}\]

where \(Q'_b(z)\) is the borehole heat extraction rate per unit depth at a depth \((z)\), \(T_b(z)\) is the borehole wall temperature, \(\bar{T}_f(z)\) is the arithmetic mean fluid temperature and \(R_b\) is the local borehole thermal resistance.

Returns

R_bfloat: Local borehole thermal resistance (in m.K/W).

update_thermal_resistances(R_fp)¶

Update the delta-circuit of thermal resistances.

This methods updates the values of the delta-circuit thermal resistances based on the provided fluid to outer pipe wall thermal resistance.

Parameters

R_fpfloat: Fluid to outer pipe wall thermal resistance (m-K/W).

visualize_pipes()¶

Plot the cross-section view of the borehole.

Returns

figfigure: Figure object (matplotlib).

class pygfunction.pipes.SingleUTube(pos, r_in, r_out, borehole, k_s, k_g, R_fp, J=2)¶

Bases: _BasePipe

Class for single U-Tube boreholes.

Contains information regarding the physical dimensions and thermal characteristics of the pipes and the grout material, as well as methods to evaluate fluid temperatures and heat extraction rates based on the work of Hellstrom 9. Internal borehole thermal resistances are evaluated using the multipole method of Claesson and Hellstrom 10.

Notes

The expected array shapes of input parameters and outputs are documented for each class method. nInlets and nOutlets are the number of inlets and outlets to the borehole, and both are equal to 1 for a single U-tube borehole. nSegments is the number of discretized segments along the borehole. nPipes is the number of pipes (i.e. the number of U-tubes) in the borehole, equal to 1. nDepths is the number of depths at which temperatures are evaluated.

The effective borehole thermal resistance is evaluated using the method of Cimmino 11. This is valid for any number of pipes.

References

9: Hellstrom, G. (1991). Ground heat storage. Thermal Analyses of Duct Storage Systems I: Theory. PhD Thesis. University of Lund, Department of Mathematical Physics. Lund, Sweden.
10: Claesson, J., & Hellstrom, G. (2011). Multipole method to calculate borehole thermal resistances in a borehole heat exchanger. HVAC&R Research, 17(6), 895-911.
11: Cimmino, M. (2019). Semi-analytical method for g-function calculation of bore fields with series- and parallel-connected boreholes. Science and Technology for the Built Environment, 25 (8), 1007-1022.

Attributes

poslist of tuples: Position (x, y) (in meters) of the pipes inside the borehole.
r_infloat: Inner radius (in meters) of the U-Tube pipes.
r_outfloat: Outer radius (in meters) of the U-Tube pipes.
boreholeBorehole object: Borehole class object of the borehole containing the U-Tube.
k_sfloat: Soil thermal conductivity (in W/m-K).
k_gfloat: Grout thermal conductivity (in W/m-K).
R_fpfloat: Fluid to outer pipe wall thermal resistance (m-K/W).
Jint, optional: Number of multipoles per pipe to evaluate the thermal resistances. Default is 2.
nPipesint: Number of U-Tubes, equals to 1.
nInletsint: Total number of pipe inlets, equals to 1.
nOutletsint: Total number of pipe outlets, equals to 1.

coefficients_borehole_heat_extraction_rate(m_flow_borehole, cp_f, nSegments, segment_ratios=None)¶

Build coefficient matrices to evaluate heat extraction rates.

Returns coefficients for the relation:

\[\mathbf{Q_b} = \mathbf{a_{in}} \mathbf{T_{f,in}} + \mathbf{a_{b}} \mathbf{T_b}\]

Parameters

m_flow_boreholefloat or (nInlets,) array: Inlet mass flow rate (in kg/s) into the borehole.
cp_ffloat or (nInlets,) array: Fluid specific isobaric heat capacity (in J/kg.degC).
nSegmentsint: Number of borehole segments.
segment_ratios(nSegments,) array, optional: Ratio of the borehole length represented by each segment. The sum of ratios must be equal to 1. If segment_ratios==None, segments of equal lengths are considered. Default is None.

Returns

a_in(nSegments, nInlets,) array: Array of coefficients for inlet fluid temperature.
a_b(nSegments, nSegments,) array: Array of coefficients for borehole wall temperatures.

coefficients_fluid_heat_extraction_rate(m_flow_borehole, cp_f, nSegments, segment_ratios=None)¶

Build coefficient matrices to evaluate heat extraction rates.

Returns coefficients for the relation:

\[\mathbf{Q_f} = \mathbf{a_{in}} \mathbf{T_{f,in}} + \mathbf{a_{b}} \mathbf{T_b}\]

Parameters

m_flow_boreholefloat or (nInlets,) array: Inlet mass flow rate (in kg/s) into the borehole.
cp_ffloat or (nInlets,) array: Fluid specific isobaric heat capacity (in J/kg.degC).
nSegmentsint: Number of borehole segments.
segment_ratios(nSegments,) array, optional: Ratio of the borehole length represented by each segment. The sum of ratios must be equal to 1. If segment_ratios==None, segments of equal lengths are considered. Default is None.

Returns

a_in(nOutlets, nInlets,) array: Array of coefficients for inlet fluid temperature.
a_b(nOutlets, nSegments,) array: Array of coefficients for borehole wall temperatures.

coefficients_inlet_temperature(m_flow_borehole, cp_f, nSegments, segment_ratios=None)¶

Build coefficient matrices to evaluate inlet fluid temperature.

Returns coefficients for the relation:

\[\mathbf{T_{f,in}} = \mathbf{a_{q,f}} \mathbf{Q_{f}} + \mathbf{a_{b}} \mathbf{T_b}\]

Parameters

m_flow_boreholefloat or (nInlets,) array: Inlet mass flow rates (in kg/s) into the borehole.
cp_ffloat or (nInlets,) array: Fluid specific isobaric heat capacity (in J/kg.degC).
nSegmentsint: Number of borehole segments.
segment_ratios(nSegments,) array, optional: Ratio of the borehole length represented by each segment. The sum of ratios must be equal to 1. If segment_ratios==None, segments of equal lengths are considered. Default is None.

Returns

a_qf(nOutlets, nInlets,) array: Array of coefficients for inlet fluid temperature.
a_b(nOutlets, nSegments,) array: Array of coefficients for borehole wall temperatures.

coefficients_outlet_temperature(m_flow_borehole, cp_f, nSegments, segment_ratios=None)¶

Build coefficient matrices to evaluate outlet fluid temperature.

Returns coefficients for the relation:

\[\mathbf{T_{f,out}} = \mathbf{a_{in}} \mathbf{T_{f,in}} + \mathbf{a_{b}} \mathbf{T_b}\]

Parameters

m_flow_boreholefloat or (nInlets,) array: Inlet mass flow rates (in kg/s) into the borehole.
cp_ffloat or (nInlets,) array: Fluid specific isobaric heat capacity (in J/kg.degC).
nSegmentsint: Number of borehole segments.
segment_ratios(nSegments,) array, optional: Ratio of the borehole length represented by each segment. The sum of ratios must be equal to 1. If segment_ratios==None, segments of equal lengths are considered. Default is None.

Returns

a_in(nOutlets, nInlets,) array: Array of coefficients for inlet fluid temperature.
a_b(nOutlets, nSegments,) array: Array of coefficients for borehole wall temperatures.

coefficients_temperature(z, m_flow_borehole, cp_f, nSegments, segment_ratios=None)¶

Build coefficient matrices to evaluate fluid temperatures at a depth (z).

Returns coefficients for the relation:

\[\mathbf{T_f}(z) = \mathbf{a_{in}} \mathbf{T_{f,in}} + \mathbf{a_{b}} \mathbf{T_b}\]

Parameters

zfloat or (nDepths,) array: Depths (in meters) to evaluate the fluid temperature coefficients.
m_flow_boreholefloat or (nInlets,) array: Inlet mass flow rate (in kg/s) into the borehole.
cp_ffloat or (nInlets,) array: Fluid specific isobaric heat capacity (in J/kg.degC).
nSegmentsint: Number of borehole segments.
segment_ratios(nSegments,) array, optional: Ratio of the borehole length represented by each segment. The sum of ratios must be equal to 1. If segment_ratios==None, segments of equal lengths are considered. Default is None.

Returns

a_in
(2*nPipes, nInlets,) array, or (nDepths, 2*nPipes, nInlets,) array: Array of coefficients for inlet fluid temperature.
a_b
(2*nPipes, nSegments,) array, or (nDepths, 2*nPipes, nSegments,) array: Array of coefficients for borehole wall temperatures.

effective_borehole_thermal_resistance(m_flow_borehole, cp_f)¶

Evaluate the effective borehole thermal resistance, defined by:

\[ \begin{align}\begin{aligned}\frac{Q_b}{H} = \frac{T^*_b - \bar{T}_f}{R^*_b}\\\bar{T}_f = \frac{1}{2}(T_{f,in} + T_{f,out})\end{aligned}\end{align} \]

where \(Q_b\) is the borehole heat extraction rate (in Watts), \(H\) is the borehole length, \(T^*_b\) is the effective borehole wall temperature, \(R^*_b\) is the effective borehole thermal resistance, \(T_{f,in}\) is the inlet fluid temperature, and \(T_{f,out}\) is the outlet fluid temperature.

Parameters

m_flow_boreholefloat: Fluid mass flow rate (in kg/s) into the borehole.
cp_ffloat: Fluid specific isobaric heat capacity (in J/kg.K)

Returns

R_bfloat: Effective borehole thermal resistance (in m.K/W).

get_borehole_heat_extraction_rate(T_f_in, T_b, m_flow_borehole, cp_f, segment_ratios=None)¶

Returns the heat extraction rates of the borehole.

Parameters

T_f_infloat or (nInlets,) array: Inlet fluid temperatures (in Celsius).
T_bfloat or (nSegments,) array: Borehole wall temperatures (in Celsius).
m_flow_boreholefloat or (nInlets,) array: Inlet mass flow rates (in kg/s) into the borehole.
cp_ffloat or (nInlets,) array: Fluid specific isobaric heat capacity (in J/kg.degC).
segment_ratios(nSegments,) array, optional: Ratio of the borehole length represented by each segment. The sum of ratios must be equal to 1. If segment_ratios==None, segments of equal lengths are considered. Default is None.

Returns

Q_bfloat or (nSegments,) array: Heat extraction rates along each borehole segment (in Watts). The returned type corresponds to the type of the parameter T_b.

get_fluid_heat_extraction_rate(T_f_in, T_b, m_flow_borehole, cp_f, segment_ratios=None)¶

Returns the heat extraction rates of the borehole.

Parameters

T_f_infloat or (nInlets,) array: Inlet fluid temperatures (in Celsius).
T_bfloat or (nSegments,) array: Borehole wall temperatures (in Celsius).
m_flow_boreholefloat or (nInlets,) array: Inlet mass flow rates (in kg/s) into the borehole.
cp_ffloat or (nInlets,) array: Fluid specific isobaric heat capacity (in J/kg.degC).
segment_ratios(nSegments,) array, optional: Ratio of the borehole length represented by each segment. The sum of ratios must be equal to 1. If segment_ratios==None, segments of equal lengths are considered. Default is None.

Returns

Q_ffloat or (nOutlets,) array: Heat extraction rates from each fluid circuit (in Watts). The returned type corresponds to the type of the parameter T_f_in.

get_inlet_temperature(Q_f, T_b, m_flow_borehole, cp_f, segment_ratios=None)¶

Returns the inlet fluid temperatures of the borehole.

Parameters

Q_ffloat or (nInlets,) array: Heat extraction from the fluid circuits (in Watts).
T_bfloat or (nSegments,) array: Borehole wall temperatures (in Celsius).
m_flow_boreholefloat or (nInlets,) array: Inlet mass flow rates (in kg/s) into the borehole.
cp_ffloat or (nInlets,) array: Fluid specific isobaric heat capacity (in J/kg.degC).
segment_ratios(nSegments,) array, optional: Ratio of the borehole length represented by each segment. The sum of ratios must be equal to 1. If segment_ratios==None, segments of equal lengths are considered. Default is None.

Returns

T_infloat or (nOutlets,) array: Inlet fluid temperatures (in Celsius) into each inlet pipe. The returned type corresponds to the type of the parameter Q_f.

get_outlet_temperature(T_f_in, T_b, m_flow_borehole, cp_f, segment_ratios=None)¶

Returns the outlet fluid temperatures of the borehole.

Parameters

T_f_infloat or (nInlets,) array: Inlet fluid temperatures (in Celsius).
T_bfloat or (nSegments,) array: Borehole wall temperatures (in Celsius).
m_flow_boreholefloat or (nInlets,) array: Inlet mass flow rates (in kg/s) into the borehole.
cp_ffloat or (nInlets,) array: Fluid specific isobaric heat capacity (in J/kg.degC).
segment_ratios(nSegments,) array, optional: Ratio of the borehole length represented by each segment. The sum of ratios must be equal to 1. If segment_ratios==None, segments of equal lengths are considered. Default is None.

Returns

T_f_outfloat or (nOutlets,) array: Outlet fluid temperatures (in Celsius) from each outlet pipe. The returned type corresponds to the type of the parameter T_f_in.

get_temperature(z, T_f_in, T_b, m_flow_borehole, cp_f, segment_ratios=None)¶

Returns the fluid temperatures of the borehole at a depth (z).

Parameters

zfloat or (nDepths,) array: Depths (in meters) to evaluate the fluid temperatures.
T_f_infloat or (nInlets,) array: Inlet fluid temperatures (in Celsius).
T_bfloat or (nSegments,) array: Borehole wall temperatures (in Celsius).
m_flow_boreholefloat or (nInlets,) array: Inlet mass flow rates (in kg/s) into the borehole.
cp_ffloat or (nInlets,) array: Fluid specific isobaric heat capacity (in J/kg.degC).
segment_ratios(nSegments,) array, optional: Ratio of the borehole length represented by each segment. The sum of ratios must be equal to 1. If segment_ratios==None, segments of equal lengths are considered. Default is None.

Returns

T_f(2*nPipes,) or (nDepths, 2*nPipes,) array: Fluid temperature (in Celsius) in each pipe. The returned shape depends on the type of the parameter z.

get_total_heat_extraction_rate(T_f_in, T_b, m_flow_borehole, cp_f, segment_ratios=None)¶

Returns the total heat extraction rate of the borehole.

Parameters

T_f_infloat or (nInlets,) array: Inlet fluid temperatures (in Celsius).
T_bfloat or (nSegments,) array: Borehole wall temperatures (in Celsius).
m_flow_boreholefloat or (nInlets,) array: Inlet mass flow rates (in kg/s) into the borehole.
cp_ffloat or (nInlets,) array: Fluid specific isobaric heat capacity (in J/kg.degC).
segment_ratios(nSegments,) array, optional: Ratio of the borehole length represented by each segment. The sum of ratios must be equal to 1. If segment_ratios==None, segments of equal lengths are considered. Default is None.

Returns

Q_tfloat: Total net heat extraction rate of the borehole (in Watts).

local_borehole_thermal_resistance()¶

Evaluate the local (cross-sectional) borehole thermal resistance, defined by:

\[Q'_b(z) = \frac{T_b(z) - \bar{T}_f(z)}{R_b}\]

where \(Q'_b(z)\) is the borehole heat extraction rate per unit depth at a depth \((z)\), \(T_b(z)\) is the borehole wall temperature, \(\bar{T}_f(z)\) is the arithmetic mean fluid temperature and \(R_b\) is the local borehole thermal resistance.

Returns

R_bfloat: Local borehole thermal resistance (in m.K/W).

update_thermal_resistances(R_fp)¶

Update the delta-circuit of thermal resistances.

This methods updates the values of the delta-circuit thermal resistances based on the provided fluid to outer pipe wall thermal resistance.

Parameters

R_fpfloat: Fluid to outer pipe wall thermal resistance (m-K/W).

visualize_pipes()¶

Plot the cross-section view of the borehole.

Returns

figfigure: Figure object (matplotlib).

pygfunction.pipes.borehole_thermal_resistance(pipe, m_flow_borehole, cp_f)¶

Evaluate the effective borehole thermal resistance, defined by:

\[ \begin{align}\begin{aligned}\frac{Q_b}{H} = \frac{T^*_b - \bar{T}_f}{R^*_b}\\\bar{T}_f = \frac{1}{2}(T_{f,in} + T_{f,out})\end{aligned}\end{align} \]

where \(Q_b\) is the borehole heat extraction rate (in Watts), \(H\) is the borehole length, \(T^*_b\) is the effective borehole wall temperature, \(R^*_b\) is the effective borehole thermal resistance, \(T_{f,in}\) is the inlet fluid temperature, and \(T_{f,out}\) is the outlet fluid temperature.

Parameters

pipepipe object: Model for pipes inside the borehole.
m_flow_boreholefloat: Fluid mass flow rate (in kg/s) into the borehole.
cp_ffloat: Fluid specific isobaric heat capacity (in J/kg.K)

Returns

R_bfloat: Effective borehole thermal resistance (m.K/W).

Notes

The effective borehole thermal resistance is evaluated using the method of Cimmino 12. This is valid for any number of pipes.

References

12: Cimmino, M. (2019). Semi-analytical method for g-function calculation of bore fields with series- and parallel-connected boreholes. Science and Technology for the Built Environment, 25 (8), 1007-1022.

pygfunction.pipes.conduction_thermal_resistance_circular_pipe(r_in, r_out, k_p)¶

Evaluate the conduction thermal resistance for circular pipes.

Parameters

r_infloat: Inner radius of the pipes (in meters).
r_outfloat: Outer radius of the pipes (in meters).
k_pfloat: Pipe thermal conductivity (in W/m-K).

Returns

R_pfloat: Conduction thermal resistance (in m-K/W).

pygfunction.pipes.convective_heat_transfer_coefficient_circular_pipe(m_flow_pipe, r_in, mu_f, rho_f, k_f, cp_f, epsilon)¶

Evaluate the convective heat transfer coefficient for circular pipes.

The Nusselt number must first be determined to find the convection coefficient. Determination of the Nusselt number in turbulent flow is done by calling _Nusselt_number_turbulent_flow(). An analytical solution for constant pipe wall surface temperature is used for laminar flow.

Since _Nusselt_number_turbulent_flow() is only valid for Re > 3000. and to avoid dicontinuities in the values of the convective heat transfer coefficient near the onset of the turbulence region (approximately Re = 2300.), linear interpolation is used over the range 2300 < Re < 4000 for the evaluation of the Nusselt number.

This approach was verified by Gnielinski (2013) 13.

Parameters

m_flow_pipefloat: Fluid mass flow rate (in kg/s) into the pipe.
r_infloat: Inner radius of the pipe (in meters).
mu_ffloat: Fluid dynamic viscosity (in kg/m-s).
rho_ffloat: Fluid density (in kg/m3).
k_ffloat: Fluid thermal conductivity (in W/m-K).
cp_ffloat: Fluid specific heat capacity (in J/kg-K).
epsilonfloat: Pipe roughness (in meters).

Returns

h_fluidfloat: Convective heat transfer coefficient (in W/m2-K).

References

13(1,2): Gnielinski, V. (2013). On heat transfer in tubes. International Journal of Heat and Mass Transfer, 63, 134–140. https://doi.org/10.1016/j.ijheatmasstransfer.2013.04.015

pygfunction.pipes.convective_heat_transfer_coefficient_concentric_annulus(m_flow_pipe, r_a_in, r_a_out, mu_f, rho_f, k_f, cp_f, epsilon)¶

Evaluate the inner and outer convective heat transfer coefficient for the annulus region of a concentric pipe.

Grundmann (2016) referenced Hellström (1991) 17 in the discussion about inner and outer convection coefficients in an annulus region of a concentric pipe arrangement.

The following is valid for \(Re < 2300\) and \(0.1 \leq Pr \leq 1000\) :

\[\text{Nu}_{a,in} = 3.66 + 1.2(r^*)^{-0.8}\]

\[\text{Nu}_{a,out} = 3.66 + 1.2(r^*)^{0.5}\]

where \(r^* = r_{a,in} / r_{a,out}\) is the ratio of the inner over the outer annulus radius (Çengel and Ghajar 2015, pg. 476).

Cengel and Ghajar (2015) 15 state that inner and outer Nusselt numbers are approximately equivalent for turbulent flow. They additionally state that Gnielinski _Nusselt_number_turbulent_flow() can be used for turbulent flow. Linear interpolation is used over the range 2300 < Re < 4000 for the evaluation of the Nusselt number, as proposed by Gnielinski (2013) 13.

Parameters

m_flow_pipe: float: Fluid mass flow rate (in kg/s) into the pipe.
r_a_in: float: Pipe annulus inner radius (in meters).
r_a_out: float: Pipe annulus outer radius (in meters).
mu_ffloat: Fluid dynamic viscosity (in kg/m-s).
rho_ffloat: Fluid density (in kg/m3).
k_ffloat: Fluid thermal conductivity (in W/m-K).
cp_ffloat: Fluid specific heat capacity (in J/kg-K).
epsilonfloat: Pipe roughness (in meters).

Returns

h_fluid_a_in: float: Convective heat transfer coefficient of the inner pipe annulus region (in W/m2-K).
h_fluid_a_out: float: Convective heat transfer coefficient of the outer pipe annulus region (in W/m2-K).

References

14: Grundmann, R. (2016) Improved design methods for ground heat exchangers. Oklahoma State University, M.S. Thesis.
15: Çengel, Y.A., & Ghajar, A.J. (2015). Heat and mass transfer: fundamentals & applications (Fifth edition.). McGraw-Hill.

pygfunction.pipes.fluid_friction_factor_circular_pipe(m_flow_pipe, r_in, mu_f, rho_f, epsilon, tol=1e-06)¶

Evaluate the Darcy-Weisbach friction factor.

Parameters

m_flow_pipefloat: Fluid mass flow rate (in kg/s) into the pipe.
r_infloat: Inner radius of the pipes (in meters).
mu_ffloat: Fluid dynamic viscosity (in kg/m-s).
rho_ffloat: Fluid density (in kg/m3).
epsilonfloat: Pipe roughness (in meters).
tolfloat: Relative convergence tolerance on Darcy friction factor. Default is 1.0e-6.

Returns

fDarcyfloat: Darcy friction factor.

pygfunction.pipes.multipole(pos, r_out, r_b, k_s, k_g, R_fp, T_b, q_p, J, x_T=array([], dtype=float64), y_T=array([], dtype=float64), eps=1e-05, it_max=100)¶

Multipole method to calculate borehole thermal resistances in a borehole heat exchanger.

Adapted from the work of Claesson and Hellstrom 16.

Parameters

poslist: List of positions (x,y) (in meters) of pipes around the center of the borehole.
r_outfloat or array: Outer radius of the pipes (in meters).
r_bfloat: Borehole radius (in meters).
k_sfloat: Soil thermal conductivity (in W/m-K).
k_gfloat: Grout thermal conductivity (in W/m-K).
R_fpfloat or array: Fluid-to-outer-pipe-wall thermal resistance (in m-K/W).
Jint: Number of multipoles per pipe to evaluate the thermal resistances. J=1 or J=2 usually gives sufficient accuracy. J=0 corresponds to the line source approximation.
q_parray: Thermal energy flows (in W/m) from pipes.
T_bfloat: Average borehole wall temperature (in degC).
epsfloat, optional: Iteration relative accuracy. Default is 1e-5.
it_maxint, optional: Maximum number of iterations. Default is 100.
x_Tarray, optional: x-coordinates (in meters) to calculate temperatures. Default is np.empty(0).
y_Tarray, optional: y-coordinates (in meters) to calculate temperatures. Default is np.empty(0).

Returns

T_farray: Fluid temperatures (in degC) in the pipes.
Tarray: Requested temperatures (in degC).
itint: Total number of iterations
eps_maxfloat: Maximum error.

References

16: Claesson, J., & Hellstrom, G. (2011). Multipole method to calculate borehole thermal resistances in a borehole heat exchanger. HVAC&R Research, 17(6), 895-911.

pygfunction.pipes.thermal_resistances(pos, r_out, r_b, k_s, k_g, R_fp, J=2)¶

Evaluate thermal resistances and delta-circuit thermal resistances.

This function evaluates the thermal resistances and delta-circuit thermal resistances between pipes in a borehole using the multipole method 18. Thermal resistances are defined by:

\[\mathbf{T_f} - T_b = \mathbf{R} \cdot \mathbf{Q_{p}}\]

Delta-circuit thermal resistances are defined by:

\[ \begin{align}\begin{aligned}q_{p,i,j} = \frac{T_{f,i} - T_{f,j}}{R^\Delta_{i,j}}\\q_{p,i,i} = \frac{T_{f,i} - T_b}{R^\Delta_{i,i}}\end{aligned}\end{align} \]

Parameters

poslist: List of positions (x,y) (in meters) of pipes around the center of the borehole.
r_outfloat or array: Outer radius of the pipes (in meters).
r_bfloat: Borehole radius (in meters).
k_sfloat: Soil thermal conductivity (in W/m-K).
k_gfloat: Grout thermal conductivity (in W/m-K).
R_fpfloat or array: Fluid-to-outer-pipe-wall thermal resistance (in m-K/W).
Jint, optional: Number of multipoles per pipe to evaluate the thermal resistances. J=1 or J=2 usually gives sufficient accuracy. J=0 corresponds to the line source approximation 17. Default is 2.

Returns

Rarray: Thermal resistances (in m-K/W).
Rdarray: Delta-circuit thermal resistances (in m-K/W).

References

17(1,2): Hellstrom, G. (1991). Ground heat storage. Thermal Analyses of Duct Storage Systems I: Theory. PhD Thesis. University of Lund, Department of Mathematical Physics. Lund, Sweden.
18: Claesson, J., & Hellstrom, G. (2011). Multipole method to calculate borehole thermal resistances in a borehole heat exchanger. HVAC&R Research, 17(6), 895-911.

Examples

>>> pos = [(-0.06, 0.), (0.06, 0.)]
>>> R, Rd = gt.pipes.thermal_resistances(pos, 0.01, 0.075, 2., 1., 0.1,
                                         J=0)
R = [[ 0.36648149, -0.04855895],
     [-0.04855895,  0.36648149]]
Rd = [[ 0.31792254, -2.71733044],
      [-2.71733044,  0.31792254]]