Buildings.Fluid.Geothermal.Borefields.BaseClasses.Boreholes.BaseClasses.Functions
Package with functions for evaluation of borehole thermal resistances
Information
This package contains functions to evaluate borehole internal resistances used by models in Buildings.Fluid.Geothermal.Borefields.BaseClasses.Boreholes.BaseClasses.
Extends from Modelica.Icons.VariantsPackage (Icon for package containing variants).
Package Content
| Name | Description |
|---|---|
 convectionResistanceCircularPipe
|
Thermal resistance from the fluid in pipes and the grout zones (Bauer et al. 2011) |
 internalResistancesOneUTube
|
Thermal resistances for single U-tube, according to Bauer et al. (2011) |
| internalResistancesTwoUTube
|
Thermal resistances for double U-tube, according to Bauer et al (2011) |
| multipoleFluidTemperature
|
Fluid temperatures from multipole solution |
| multipoleFmk
|
Complex matrix F_mk from Claesson and Hellstrom (2011) |
| multipoleThermalResistances
|
Thermal resistances from multipole solution |
| partialInternalResistances
|
Partial model for borehole resistance calculation |
 Validation
|
Models to validate borehole thermal resistances functions |
Buildings.Fluid.Geothermal.Borefields.BaseClasses.Boreholes.BaseClasses.Functions.convectionResistanceCircularPipe
Thermal resistance from the fluid in pipes and the grout zones (Bauer et al. 2011)
Information
This model computes the convection resistance in the pipes of a borehole segment with heigth hSeg using correlations suggested by Bergman et al. (2011).
If the flow is laminar (Re ≤ 2300, with Re being the Reynolds number of the flow), the Nusselt number of the flow is assumed to be constant at 3.66. If the flow is turbulent (Re > 2300), the correlation of Dittus-Boelter is used to find the convection heat transfer coefficient as
Nu = 0.023 Re0.8 Prn,
where Nu is the Nusselt number and Pr is the Prandlt number. A value of n=0.35 is used, as the reference uses n=0.4 for heating and n=0.3 for cooling. To ensure that the function is continuously differentiable, a smooth transition between the laminar and turbulent values is created for the range 2300 < Re < 2400.
References
Bergman, T. L., Incropera, F. P., DeWitt, D. P., & Lavine, A. S. (2011). Fundamentals of heat and mass transfer (7th ed.). New York: John Wiley & Sons.
Inputs
| Type | Name | Default | Description |
|---|---|---|---|
| Height | hSeg | Height of the element [m] | |
| Radius | rTub | Tube radius [m] | |
| Length | eTub | Tube thickness [m] | |
| ThermalConductivity | kMed | Thermal conductivity of the fluid [W/(m.K)] | |
| DynamicViscosity | muMed | Dynamic viscosity of the fluid [Pa.s] | |
| SpecificHeatCapacity | cpMed | Specific heat capacity of the fluid [J/(kg.K)] | |
| MassFlowRate | m_flow | Mass flow rate [kg/s] | |
| MassFlowRate | m_flow_nominal | Nominal mass flow rate [kg/s] |
Outputs
| Type | Name | Description |
|---|---|---|
| ThermalResistance | RFluPip | Convection resistance (or conduction in fluid if no mass flow) [K/W] |
Modelica definition
function convectionResistanceCircularPipe
"Thermal resistance from the fluid in pipes and the grout zones (Bauer et al. 2011)"
// Geometry of the borehole
input Modelica.SIunits.Height hSeg "Height of the element";
input Modelica.SIunits.Radius rTub "Tube radius";
input Modelica.SIunits.Length eTub "Tube thickness";
// thermal properties
input Modelica.SIunits.ThermalConductivity kMed
"Thermal conductivity of the fluid";
input Modelica.SIunits.DynamicViscosity muMed
"Dynamic viscosity of the fluid";
input Modelica.SIunits.SpecificHeatCapacity cpMed
"Specific heat capacity of the fluid";
input Modelica.SIunits.MassFlowRate m_flow "Mass flow rate";
input Modelica.SIunits.MassFlowRate m_flow_nominal "Nominal mass flow rate";
// Outputs
output Modelica.SIunits.ThermalResistance RFluPip
"Convection resistance (or conduction in fluid if no mass flow)";
protected
parameter Modelica.SIunits.Radius rTub_in = rTub - eTub
"Pipe inner radius";
Modelica.SIunits.CoefficientOfHeatTransfer h
"Convective heat transfer coefficient of the fluid";
Real k(unit="s/kg")
"Coefficient used in the computation of the convective heat transfer coefficient";
Modelica.SIunits.MassFlowRate m_flow_abs = Buildings.Utilities.Math.Functions.spliceFunction(m_flow,-m_flow,m_flow,m_flow_nominal/30);
Real Re "Reynolds number";
Real NuTurb "Nusselt at Re=2400";
Real Nu "Nusselt";
algorithm
// Convection resistance and Reynolds number
k := 2/(muMed*Modelica.Constants.pi*rTub_in);
Re := m_flow_abs*k;
if Re>=2400 then
// Turbulent, fully-developped flow in a smooth circular pipe with the
// Dittus-Boelter correlation: h = 0.023*k_f*Re*Pr/(2*rTub)
// Re = rho*v*DTub / mue_f
// = m_flow/(pi r^2) * DTub/mue_f = 2*m_flow / ( mue*pi*rTub)
Nu := 0.023*(cpMed*muMed/kMed)^(0.35)*
Buildings.Utilities.Math.Functions.regNonZeroPower(
x=Re,
n=0.8,
delta=0.01*m_flow_nominal*k);
else
// Laminar, fully-developped flow in a smooth circular pipe with uniform
// imposed temperature: Nu=3.66 for Re<=2300. For 2300<Re<2400, a smooth
// transition is created with the splice function.
NuTurb := 0.023*(cpMed*muMed/kMed)^(0.35)*(2400)^(0.8);
Nu := Buildings.Utilities.Math.Functions.spliceFunction(NuTurb,3.66,Re-(2300+2400)/2,((2300+2400)/2)-2300);
end if;
h := Nu*kMed/(2*rTub_in);
RFluPip := 1/(2*Modelica.Constants.pi*rTub_in*hSeg*h);
end convectionResistanceCircularPipe;
Buildings.Fluid.Geothermal.Borefields.BaseClasses.Boreholes.BaseClasses.Functions.internalResistancesOneUTube
Thermal resistances for single U-tube, according to Bauer et al. (2011)
Information
This model computes the different thermal resistances present in a single-U-tube borehole using the method of Bauer et al. (2011). It also computes the fluid-to-ground thermal resistance Rb and the grout-to-grout thermal resistance Ra as defined by Claesson and Hellstrom (2011) using the multipole method.
References
J. Claesson and G. Hellstrom. Multipole method to calculate borehole thermal resistances in a borehole heat exchanger. HVAC&R Research, 17(6): 895-911, 2011.
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.
Extends from Buildings.Fluid.Geothermal.Borefields.BaseClasses.Boreholes.BaseClasses.Functions.partialInternalResistances (Partial model for borehole resistance calculation).
Inputs
| Type | Name | Default | Description |
|---|---|---|---|
| Boolean | use_Rb | false | True if the value Rb should be used instead of calculated |
| Real | Rb | Borehole thermal resistance [(m.K)/W] | |
| Height | hSeg | Height of the element [m] | |
| Radius | rBor | Radius of the borehole [m] | |
| Radius | rTub | Radius of the tube [m] | |
| Length | eTub | Thickness of the tubes [m] | |
| Length | sha | Shank spacing, defined as the distance between the center of a pipe and the center of the borehole [m] | |
| ThermalConductivity | kFil | Thermal conductivity of the grout [W/(m.K)] | |
| ThermalConductivity | kSoi | Thermal conductivity of the soi [W/(m.K)] | |
| ThermalConductivity | kTub | Thermal conductivity of the tube [W/(m.K)] | |
| ThermalConductivity | kMed | Thermal conductivity of the fluid [W/(m.K)] | |
| DynamicViscosity | muMed | Dynamic viscosity of the fluid [Pa.s] | |
| SpecificHeatCapacity | cpMed | Specific heat capacity of the fluid [J/(kg.K)] | |
| MassFlowRate | m_flow_nominal | Nominal mass flow rate [kg/s] | |
| Boolean | printDebug | false | Print resistances values in log for debug purposes. |
Outputs
| Type | Name | Description |
|---|---|---|
| Real | x | Capacity location |
| ThermalResistance | Rgb | Thermal resistance between grout zone and borehole wall [K/W] |
| ThermalResistance | Rgg | Thermal resistance between the two grout zones [K/W] |
| ThermalResistance | RCondGro | Thermal resistance between: pipe wall to capacity in grout [K/W] |
Modelica definition
function internalResistancesOneUTube
"Thermal resistances for single U-tube, according to Bauer et al. (2011)"
extends Buildings.Fluid.Geothermal.Borefields.BaseClasses.Boreholes.BaseClasses.Functions.partialInternalResistances;
// Outputs
output Modelica.SIunits.ThermalResistance Rgb
"Thermal resistance between grout zone and borehole wall";
output Modelica.SIunits.ThermalResistance Rgg
"Thermal resistance between the two grout zones";
output Modelica.SIunits.ThermalResistance RCondGro
"Thermal resistance between: pipe wall to capacity in grout";
protected
Real[2,2] RDelta(each unit="(m.K)/W") "Delta-circuit thermal resistances";
Real[2,2] R(each unit="(m.K)/W") "Internal thermal resistances";
Modelica.SIunits.Position[2] xPip = {-sha, sha} "x-Coordinates of pipes";
Modelica.SIunits.Position[2] yPip = {0., 0.} "y-Coordinates of pipes";
Modelica.SIunits.Radius[2] rPip = {rTub, rTub} "Outer radius of pipes";
Real[2] RFluPip(each unit="(m.K)/W") = {RCondPipe+RConv, RCondPipe+RConv} "Fluid to pipe wall thermal resistances";
Modelica.SIunits.ThermalResistance Rg
"Thermal resistance between outer borehole wall and one tube";
Modelica.SIunits.ThermalResistance Rar
"Thermal resistance between the two pipe outer walls";
Real Ra(unit="(m.K)/W")
"Grout-to-grout resistance (2D) as defined by Hellstrom. Interaction between the different grout parts";
algorithm
// Internal thermal resistances
(RDelta, R) :=
Buildings.Fluid.Geothermal.Borefields.BaseClasses.Boreholes.BaseClasses.Functions.multipoleThermalResistances(
2, 3, xPip, yPip, rBor, rPip, kFil, kSoi, RFluPip);
// Rb and Ra
Rb_multipole := 1./(1./RDelta[1,1] + 1./RDelta[2,2]);
Rb_internal := if use_Rb then Rb else Rb_multipole;
// The short-circuit resistance in weigthed by the ratio between the used
// value of Rb and the theoretical value
Ra := (R[1,1] + R[2,2] - 2*R[1,2])*Rb_internal/Rb_multipole;
// Conversion of Rb (resp. Ra) to Rg (resp. Rar) of Bauer:
Rg :=(2*Rb_internal-RCondPipe-RConv)/hSeg;
Rar :=(Ra-2*(RCondPipe + RConv))/hSeg;
// If any of the internal delta-circuit resistances is negative, then
// the location (x) of the thermal capacity is set to zero to limit
// instabilities in the calculations. Otherwise, calculations follow the
// method of Bauer et al. (2011).
if (RDelta[1,2] < 0) then
//Thermal resistance between the grout zone and borehole wall
Rgb := Rg;
//Conduction resistance in grout from pipe wall to capacity in grout
RCondGro := RCondPipe/hSeg;
//Thermal resistance between the two grout zones
Rgg := 2*Rgb*Rar/(2*Rgb - Rar);
test := true;
else
// ********** Resistances and capacity location according to Bauer **********
while test == false and i <= 10 loop
// Capacity location (with correction factor in case that the test is
// negative)
x := Modelica.Math.log(sqrt(rBor^2 + 2*rTub^2)/(2*rTub))/
Modelica.Math.log(rBor/(sqrt(2)*rTub))*((15 - i + 1)/15);
//Thermal resistance between the grout zone and borehole wall
Rgb := (1 - x)*Rg;
//Conduction resistance in grout from pipe wall to capacity in grout
RCondGro := x*Rg + RCondPipe/hSeg;
//Thermal resistance between the two grout zones
Rgg := 2*Rgb*(Rar - 2*x*Rg)/(2*Rgb - Rar + 2*x*Rg);
// Thermodynamic test to check if negative R values make sense. If not,
// decrease x-value.
test := ((1/Rgg + 1/2/Rgb)^(-1) > 0);
i := i + 1;
end while;
end if;
assert(test,
"Maximum number of iterations exceeded. Check the borehole geometry.
The tubes may be too close to the borehole wall.
Input to the function
Buildings.Fluid.HeatExchangers.Boreholes.BaseClasses.singleUTubeResistances
is
hSeg = " + String(hSeg) + " m
rBor = " + String(rBor) + " m
rTub = " + String(rTub) + " m
eTub = " + String(eTub) + " m
kSoi = " + String(kSoi) + " W/m/K
kFil = " + String(kFil) + " W/m/K
kTub = " + String(kTub) + " W/m/K
i = " + String(i) + "
Computed x = " + String(x) + " K/W
Rgb = " + String(Rgb) + " K/W
Rgg = " + String(Rgg) + " K/W");
if printDebug then
Modelica.Utilities.Streams.print("
Rb = " + String(Rb_internal) + " m K / W
RCondPipe = "+ String(RCondPipe) + " m K / W
RConv = " +String(RConv) +"m K / W
hSeg = " + String(hSeg) + " m
Rg = "+String(Rg) + " K / W
Ra = " + String(Ra) + " m K / W
x = " + String(x) + "
i = " + String(i));
end if;
end internalResistancesOneUTube;
Buildings.Fluid.Geothermal.Borefields.BaseClasses.Boreholes.BaseClasses.Functions.internalResistancesTwoUTube
Thermal resistances for double U-tube, according to Bauer et al (2011)
Information
This model computes the different thermal resistances present in a double U-tube borehole using the method of Bauer et al. (2011). It also computes the fluid-to-ground thermal resistance Rb and the grout-to-grout thermal resistance Ra as defined by Claesson and Hellstrom (2011) using the multipole method.
References
J. Claesson and G. Hellstrom. Multipole method to calculate borehole thermal resistances in a borehole heat exchanger. HVAC&R Research, 17(6): 895-911, 2011.
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.
Extends from Buildings.Fluid.Geothermal.Borefields.BaseClasses.Boreholes.BaseClasses.Functions.partialInternalResistances (Partial model for borehole resistance calculation).
Inputs
| Type | Name | Default | Description |
|---|---|---|---|
| Boolean | use_Rb | false | True if the value Rb should be used instead of calculated |
| Real | Rb | Borehole thermal resistance [(m.K)/W] | |
| Height | hSeg | Height of the element [m] | |
| Radius | rBor | Radius of the borehole [m] | |
| Radius | rTub | Radius of the tube [m] | |
| Length | eTub | Thickness of the tubes [m] | |
| Length | sha | Shank spacing, defined as the distance between the center of a pipe and the center of the borehole [m] | |
| ThermalConductivity | kFil | Thermal conductivity of the grout [W/(m.K)] | |
| ThermalConductivity | kSoi | Thermal conductivity of the soi [W/(m.K)] | |
| ThermalConductivity | kTub | Thermal conductivity of the tube [W/(m.K)] | |
| ThermalConductivity | kMed | Thermal conductivity of the fluid [W/(m.K)] | |
| DynamicViscosity | muMed | Dynamic viscosity of the fluid [Pa.s] | |
| SpecificHeatCapacity | cpMed | Specific heat capacity of the fluid [J/(kg.K)] | |
| MassFlowRate | m_flow_nominal | Nominal mass flow rate [kg/s] | |
| Boolean | printDebug | false | Print resistances values in log for debug purposes. |
Outputs
| Type | Name | Description |
|---|---|---|
| Real | x | Capacity location |
| ThermalResistance | Rgb | Thermal resistance between a grout capacity and the borehole wall, as defined by Bauer et al (2010) [K/W] |
| ThermalResistance | Rgg1 | Thermal resistance between two neightbouring grout capacities, as defined by Bauer et al (2010) [K/W] |
| ThermalResistance | Rgg2 | Thermal resistance between two grout capacities opposite to each other, as defined by Bauer et al (2010) [K/W] |
| ThermalResistance | RCondGro | Thermal resistance between a pipe wall and the grout capacity, as defined by Bauer et al (2010) [K/W] |
Modelica definition
function internalResistancesTwoUTube
"Thermal resistances for double U-tube, according to Bauer et al (2011)"
extends Buildings.Fluid.Geothermal.Borefields.BaseClasses.Boreholes.BaseClasses.Functions.partialInternalResistances;
// Outputs
output Modelica.SIunits.ThermalResistance Rgb
"Thermal resistance between a grout capacity and the borehole wall, as defined by Bauer et al (2010)";
output Modelica.SIunits.ThermalResistance Rgg1
"Thermal resistance between two neightbouring grout capacities, as defined by Bauer et al (2010)";
output Modelica.SIunits.ThermalResistance Rgg2
"Thermal resistance between two grout capacities opposite to each other, as defined by Bauer et al (2010)";
output Modelica.SIunits.ThermalResistance RCondGro
"Thermal resistance between a pipe wall and the grout capacity, as defined by Bauer et al (2010)";
protected
Real[4,4] RDelta(each unit="(m.K)/W") "Delta-circuit thermal resistances";
Real[4,4] R(each unit="(m.K)/W") "Internal thermal resistances";
Modelica.SIunits.Position[4] xPip = {-sha, sha, 0., 0.} "x-Coordinates of pipes";
Modelica.SIunits.Position[4] yPip = {0., 0., -sha, sha} "y-Coordinates of pipes";
Modelica.SIunits.Radius[4] rPip = {rTub, rTub, rTub, rTub} "Outer radius of pipes";
Real[4] RFluPip(each unit="(m.K)/W") = {RCondPipe+RConv, RCondPipe+RConv, RCondPipe+RConv, RCondPipe+RConv} "Fluid to pipe wall thermal resistances";
Real Ra( unit="(m.K)/W")
"Grout-to-grout resistance (2D) as defined by Hellstrom. Interaction between the different grout parts";
Modelica.SIunits.ThermalResistance Rg
"Thermal resistance between outer borehole wall and one tube";
Modelica.SIunits.ThermalResistance Rar1
"Thermal resistance between the two closest pipe outer walls";
Modelica.SIunits.ThermalResistance Rar2
"Thermal resistance between the two farthest pipe outer walls";
algorithm
// Internal thermal resistances
(RDelta, R) :=
Buildings.Fluid.Geothermal.Borefields.BaseClasses.Boreholes.BaseClasses.Functions.multipoleThermalResistances(
4, 3, xPip, yPip, rBor, rPip, kFil, kSoi, RFluPip);
// Rb and Ra
Rb_multipole := 1./(1./RDelta[1,1] + 1./RDelta[2,2] + 1./RDelta[3,3] + 1./RDelta[4,4]);
Rb_internal := if use_Rb then Rb else Rb_multipole;
// The short-circuit resistance in weigthed by the ratio between the used
// value of Rb and the theoretical value
Ra := (R[1,1] + R[3,3] - 2*R[1,3])*Rb_internal/Rb_multipole;
// ------ Calculation according to Bauer et al. (2010)
Rg := (4*Rb_internal - RCondPipe - RConv)/hSeg;
Rar1 := ((2 + sqrt(2))*Rg*hSeg*(Ra - RCondPipe)/(Rg*hSeg + Ra - RCondPipe))/hSeg;
Rar2 := sqrt(2)*Rar1;
// If any of the internal delta-circuit resistances is negative, then
// the location (x) of the thermal capacity is set to zero to limit
// instabilities in the calculations. Otherwise, calculations follow the
// method of Bauer et al. (2011).
if (RDelta[1,2] < 0) or (RDelta[1,3] < 0) then
//Thermal resistance between the grout zone and borehole wall
Rgb := Rg;
//Conduction resistance in grout from pipe wall to capacity in grout
RCondGro := RCondPipe/hSeg;
//Thermal resistance between the two grout zones
Rgg1 := 2*Rgb*(Rar1)/(2*Rgb - Rar1);
Rgg2 := 2*Rgb*(Rar2)/(2*Rgb - Rar2);
test := true;
else
// ********** Resistances and capacity location according to Bauer **********
while test == false and i <= 16 loop
// Capacity location (with correction factor in case that the test is negative)
x := Modelica.Math.log(sqrt(rBor^2 + 4*rTub^2)/(2*sqrt(2)*rTub))/
Modelica.Math.log(rBor/(2*rTub))*((15 - i + 1)/15);
//Thermal resistance between the grout zone and borehole wall
Rgb := (1 - x)*Rg;
//Conduction resistance in grout from pipe wall to capacity in grout
RCondGro := x*Rg + RCondPipe/hSeg;
//Thermal resistance between the two grout zones
Rgg1 := 2*Rgb*(Rar1 - 2*x*Rg)/(2*Rgb - Rar1 + 2*x*Rg);
Rgg2 := 2*Rgb*(Rar2 - 2*x*Rg)/(2*Rgb - Rar2 + 2*x*Rg);
// Thermodynamic test to check if negative R values make sense. If not, decrease x-value.
test := (((1/Rgg1 + 1/2/Rgb)^(-1) > 0) and ((1/Rgg2 + 1/2/Rgb)^(-1) > 0));
i := i + 1;
end while;
end if;
assert(test,
"Maximum number of iterations exceeded. Check the borehole geometry.
The tubes may be too close to the borehole wall.
Input to the function
Buildings.Fluid.HeatExchangers.Boreholes.BaseClasses.doubleUTubeResistances
is
hSeg = " + String(hSeg) + " m
rBor = " + String(rBor) + " m
rTub = " + String(rTub) + " m
eTub = " + String(eTub) + " m
sha = " + String(sha) + " m
kSoi = " + String(kSoi) + " W/m/K
kFil = " + String(kFil) + " W/m/K
kTub = " + String(kTub) + " W/m/K
Computed x = " + String(x) + " m
Rgb = " + String(Rgb) + " K/W
Rgg1 = " + String(Rgg1) + " K/W
Rgg2 = " + String(Rgg2) + " K/W");
if printDebug then
Modelica.Utilities.Streams.print("
Rb = " + String(Rb_internal) + " m K / W
RCondPipe = "+ String(RCondPipe) + " m K / W
RConv = " +String(RConv) +"m K / W
hSeg = " + String(hSeg) + " m
Rg = "+String(Rg) + " K / W
Ra = " + String(Ra) + " m K / W
x = " + String(x) + "
i = " + String(i));
end if;
end internalResistancesTwoUTube;
Buildings.Fluid.Geothermal.Borefields.BaseClasses.Boreholes.BaseClasses.Functions.multipoleFluidTemperature
Fluid temperatures from multipole solution
Information
This model evaluates the fluid temperatures using the multipole method of Claesson and Hellstrom (2011).
References
J. Claesson and G. Hellstrom. Multipole method to calculate borehole thermal resistances in a borehole heat exchanger. HVAC&R Research, 17(6): 895-911, 2011.
Extends from Modelica.Icons.Function (Icon for functions).
Inputs
| Type | Name | Default | Description |
|---|---|---|---|
| Integer | nPip | Number of pipes | |
| Integer | J | Number of multipoles | |
| Position | xPip[nPip] | x-Coordinates of pipes [m] | |
| Position | yPip[nPip] | y-Coordinates of pipes [m] | |
| Real | QPip_flow[nPip] | QPip_flow(each unit="W/m") | Heat flow in pipes [W/m] |
| Temperature | TBor | Average borehole wall temperature [K] | |
| Radius | rBor | Borehole radius [m] | |
| Radius | rPip[nPip] | Outter radius of pipes [m] | |
| ThermalConductivity | kFil | Thermal conductivity of grouting material [W/(m.K)] | |
| ThermalConductivity | kSoi | Thermal conductivity of soil material [W/(m.K)] | |
| Real | RFluPip[nPip] | RFluPip(each unit="(m.K)/W") | Fluid to pipe wall thermal resistances [(m.K)/W] |
| Real | eps | 1.0e-5 | Iteration relative accuracy |
| Integer | it_max | 100 | Maximum number of iterations |
Outputs
| Type | Name | Description |
|---|---|---|
| Temperature | TFlu[nPip] | Fluid temperature in pipes [K] |
Modelica definition
function multipoleFluidTemperature "Fluid temperatures from multipole solution"
extends Modelica.Icons.Function;
input Integer nPip "Number of pipes";
input Integer J "Number of multipoles";
input Modelica.SIunits.Position xPip[nPip] "x-Coordinates of pipes";
input Modelica.SIunits.Position yPip[nPip] "y-Coordinates of pipes";
input Real QPip_flow[nPip](each unit="W/m") "Heat flow in pipes";
input Modelica.SIunits.Temperature TBor "Average borehole wall temperature";
input Modelica.SIunits.Radius rBor "Borehole radius";
input Modelica.SIunits.Radius rPip[nPip] "Outter radius of pipes";
input Modelica.SIunits.ThermalConductivity kFil "Thermal conductivity of grouting material";
input Modelica.SIunits.ThermalConductivity kSoi "Thermal conductivity of soil material";
input Real RFluPip[nPip](each unit="(m.K)/W") "Fluid to pipe wall thermal resistances";
input Real eps=1.0e-5 "Iteration relative accuracy";
input Integer it_max=100 "Maximum number of iterations";
output Modelica.SIunits.Temperature TFlu[nPip] "Fluid temperature in pipes";
protected
Real pikFil(unit="(m.K)/W")=1/(2*Modelica.Constants.pi*kFil) "Coefficient based on grout thermal conductivity";
Real sigma=(kFil - kSoi)/(kFil + kSoi) "Thermal conductivity ratio";
Real betaPip[nPip]=2*Modelica.Constants.pi*kFil*RFluPip "Dimensionless fluid to outter pipe wall thermal resistance";
Complex zPip_i "Position of pipe i";
Complex zPip_j "Position of pipe j";
Complex P_nj "Multipole of order j for pipe n";
Real PRea[nPip,J] "Matrix of real part of multipoles";
Real PIma[nPip,J] "Matrix of imaginary part of multipole";
Complex P_nj_new "New value of multipole of order j for pipe n";
Real PRea_new[nPip,J] "New value of real part of multipoles";
Real PIma_new[nPip,J] "New value of imaginary part of multipoles";
Complex F_mk "Coefficient F of order k of pipe m";
Real FRea[nPip,J] "Real part of matrix F_mk";
Real FIma[nPip,J] "Imaginary part of matrix F_mk";
Real R0[nPip,nPip](each unit="(m.K)/W") "Line source approximation of thermal resistances";
Complex deltaTFlu "Fluid temperature difference with line source approximation";
Real rbm "Intermediate coefficient";
Modelica.SIunits.Distance dz "Pipe to pipe distance";
Real coeff[nPip,J] "Coefficient for multiplication with matrix F_mk";
Real diff "Difference in subsequent multipole evaluations";
Real diff_max "Maximum difference in subsequent multipole evaluations";
Real diff_min "Minimum difference in subsequent multipole evaluations";
Real diff0 "Difference in subsequent multipole evaluations";
Integer it "Iteration counter";
Real eps_max "Convergence variable";
algorithm
// Thermal resistance matrix from 0th order multipole
for i in 1:nPip loop
zPip_i := Complex(xPip[i], yPip[i]);
rbm := rBor^2/(rBor^2 - Modelica.ComplexMath.'abs'(zPip_i)^2);
R0[i, i] := pikFil*(log(rBor/rPip[i]) + betaPip[i] + sigma*log(rbm));
for j in 1:nPip loop
zPip_j := Complex(xPip[j], yPip[j]);
if i <> j then
dz := Modelica.ComplexMath.'abs'(zPip_i - zPip_j);
rbm := rBor^2/Modelica.ComplexMath.'abs'(rBor^2 - zPip_j*
Modelica.ComplexMath.conj(zPip_i));
R0[i, j] := pikFil*(log(rBor/dz) + sigma*log(rbm));
end if;
end for;
end for;
// Initialize maximum error and iteration counter
eps_max := 1.0e99;
it := 0;
// Multipoles
if J > 0 then
for m in 1:nPip loop
for k in 1:J loop
coeff[m, k] := -(1 - k*betaPip[m])/(1 + k*betaPip[m]);
PRea[m, k] := 0;
PIma[m, k] := 0;
end for;
end for;
while eps_max > eps and it < it_max loop
it := it + 1;
(FRea, FIma) :=
Buildings.Fluid.Geothermal.Borefields.BaseClasses.Boreholes.BaseClasses.Functions.multipoleFmk(
nPip,
J,
QPip_flow,
PRea,
PIma,
rBor,
rPip,
xPip,
yPip,
kFil,
kSoi);
for m in 1:nPip loop
for k in 1:J loop
F_mk := Complex(FRea[m, k], FIma[m, k]);
P_nj_new := coeff[m, k]*Modelica.ComplexMath.conj(F_mk);
PRea_new[m, k] := Modelica.ComplexMath.real(P_nj_new);
PIma_new[m, k] := Modelica.ComplexMath.imag(P_nj_new);
end for;
end for;
diff_max := 0;
diff_min := 1e99;
for m in 1:nPip loop
for k in 1:J loop
P_nj := Complex(PRea[m, k], PIma[m, k]);
P_nj_new := Complex(PRea_new[m, k], PIma_new[m, k]);
diff_max := max(diff_max,
Modelica.ComplexMath.'abs'(P_nj_new - P_nj));
diff_min := min(diff_min,
Modelica.ComplexMath.'abs'(P_nj_new - P_nj));
end for;
end for;
diff := diff_max - diff_min;
if it == 1 then
diff0 :=diff;
end if;
eps_max := diff/diff0;
PRea := PRea_new;
PIma := PIma_new;
end while;
end if;
// Fluid Temperatures
TFlu := TBor .+ R0*QPip_flow;
if J > 0 then
for m in 1:nPip loop
zPip_i :=Complex(xPip[m], yPip[m]);
deltaTFlu := Complex(0, 0);
for n in 1:nPip loop
zPip_j :=Complex(xPip[n], yPip[n]);
for j in 1:J loop
P_nj := Complex(PRea[n, j], PIma[n, j]);
if n <> m then
// Second term
deltaTFlu := deltaTFlu + P_nj*(rPip[n]/(zPip_i - zPip_j))^j;
end if;
// Third term
deltaTFlu := deltaTFlu + sigma*P_nj*(rPip[n]*
Modelica.ComplexMath.conj(zPip_i)/(rBor^2 - zPip_j*
Modelica.ComplexMath.conj(zPip_i)))^j;
end for;
end for;
TFlu[m] := TFlu[m] + Modelica.ComplexMath.real(deltaTFlu);
end for;
end if;
end multipoleFluidTemperature;
Buildings.Fluid.Geothermal.Borefields.BaseClasses.Boreholes.BaseClasses.Functions.multipoleFmk
Complex matrix F_mk from Claesson and Hellstrom (2011)
Information
This model evaluates the complex coefficient matrix F_mk from Claesson and Hellstrom (2011).
References
J. Claesson and G. Hellstrom. Multipole method to calculate borehole thermal resistances in a borehole heat exchanger. HVAC&R Research, 17(6): 895-911, 2011.
Extends from Modelica.Icons.Function (Icon for functions).
Inputs
| Type | Name | Default | Description |
|---|---|---|---|
| Integer | nPip | Number of pipes | |
| Integer | J | Number of multipoles | |
| Real | QPip_flow[nPip] | QPip_flow(each unit="W/m") | Heat flow in pipes [W/m] |
| Real | PRea[nPip, J] | Multipoles (Real part) | |
| Real | PIma[nPip, J] | Multipoles (Imaginary part) | |
| Radius | rBor | Borehole radius [m] | |
| Radius | rPip[nPip] | Outter radius of pipes [m] | |
| Position | xPip[nPip] | x-Coordinates of pipes [m] | |
| Position | yPip[nPip] | y-Coordinates of pipes [m] | |
| ThermalConductivity | kFil | Thermal conductivity of grouting material [W/(m.K)] | |
| ThermalConductivity | kSoi | Thermal conductivity of soil material [W/(m.K)] |
Outputs
| Type | Name | Description |
|---|---|---|
| Real | FRea[nPip, J] | Multipole coefficients |
| Real | FIma[nPip, J] | Multipole coefficients |
Modelica definition
function multipoleFmk "Complex matrix F_mk from Claesson and Hellstrom (2011)"
extends Modelica.Icons.Function;
input Integer nPip "Number of pipes";
input Integer J "Number of multipoles";
input Real QPip_flow[nPip](each unit="W/m") "Heat flow in pipes";
input Real PRea[nPip,J] "Multipoles (Real part)";
input Real PIma[nPip,J] "Multipoles (Imaginary part)";
input Modelica.SIunits.Radius rBor "Borehole radius";
input Modelica.SIunits.Radius rPip[nPip] "Outter radius of pipes";
input Modelica.SIunits.Position xPip[nPip] "x-Coordinates of pipes";
input Modelica.SIunits.Position yPip[nPip] "y-Coordinates of pipes";
input Modelica.SIunits.ThermalConductivity kFil "Thermal conductivity of grouting material";
input Modelica.SIunits.ThermalConductivity kSoi "Thermal conductivity of soil material";
output Real FRea[nPip,J] "Multipole coefficients";
output Real FIma[nPip,J] "Multipole coefficients";
protected
Complex zPip_i "Position of pipe i";
Complex zPip_j "Position of pipe j";
Complex P_nj "Multipole of order j for pipe n";
Real pikFil(unit="(m.K)/W")=1/(2*Modelica.Constants.pi*kFil) "Coefficient based on grout thermal conductivity";
Real sigma=(kFil - kSoi)/(kFil + kSoi) "Thermal conductivity ratio";
Complex f "Intermedia value of multipole coefficient";
Integer j_pend "Maximum loop index in fourth term";
algorithm
for m in 1:nPip loop
zPip_i := Complex(xPip[m], yPip[m]);
for k in 1:J loop
f := Complex(0, 0);
for n in 1:nPip loop
zPip_j := Complex(xPip[n], yPip[n]);
// First term
if m <> n then
f := f + QPip_flow[n]*pikFil/k*(rPip[m]/(zPip_j - zPip_i))^k;
end if;
// Second term
f := f + sigma*QPip_flow[n]*pikFil/k*(rPip[m]*Modelica.ComplexMath.conj(
zPip_j)/(rBor^2 - zPip_i*Modelica.ComplexMath.conj(zPip_j)))^k;
for j in 1:J loop
P_nj := Complex(PRea[n, j], PIma[n, j]);
// Third term
if m <> n then
f := f + P_nj*Buildings.Utilities.Math.Functions.binomial(j + k - 1, j -
1)*rPip[n]^j*(-rPip[m])^k/(zPip_i - zPip_j)^(j + k);
end if;
//Fourth term
j_pend := min(k, j);
for jp in 0:j_pend loop
f := f + sigma*Modelica.ComplexMath.conj(P_nj)*
Buildings.Utilities.Math.Functions.binomial(j, jp)*
Buildings.Utilities.Math.Functions.binomial(j + k - jp - 1, j - 1)*
rPip[n]^j*rPip[m]^k*zPip_i^(j - jp)*Modelica.ComplexMath.conj(
zPip_j)^(k - jp)/(rBor^2 - zPip_i*Modelica.ComplexMath.conj(
zPip_j))^(k + j - jp);
end for;
end for;
end for;
FRea[m, k] := Modelica.ComplexMath.real(f);
FIma[m, k] := Modelica.ComplexMath.imag(f);
end for;
end for;
end multipoleFmk;
Buildings.Fluid.Geothermal.Borefields.BaseClasses.Boreholes.BaseClasses.Functions.multipoleThermalResistances
Thermal resistances from multipole solution
Information
This model evaluates the delta-circuit borehole thermal resistances using the multipole method of Claesson and Hellstrom (2011).
References
J. Claesson and G. Hellstrom. Multipole method to calculate borehole thermal resistances in a borehole heat exchanger. HVAC&R Research, 17(6): 895-911, 2011.
Extends from Modelica.Icons.Function (Icon for functions).
Inputs
| Type | Name | Default | Description |
|---|---|---|---|
| Integer | nPip | Number of pipes | |
| Integer | J | Number of multipoles | |
| Position | xPip[nPip] | x-Coordinates of pipes [m] | |
| Position | yPip[nPip] | y-Coordinates of pipes [m] | |
| Radius | rBor | Borehole radius [m] | |
| Radius | rPip[nPip] | Outter radius of pipes [m] | |
| ThermalConductivity | kFil | Thermal conductivity of grouting material [W/(m.K)] | |
| ThermalConductivity | kSoi | Thermal conductivity of soil material [W/(m.K)] | |
| Real | RFluPip[nPip] | RFluPip(each unit="(m.K)/W") | Fluid to pipe wall thermal resistances [(m.K)/W] |
| Temperature | TBor | 0 | Average borehole wall temperature [K] |
Outputs
| Type | Name | Description |
|---|---|---|
| Real | RDelta[nPip, nPip] | Delta-circuit thermal resistances [(m.K)/W] |
| Real | R[nPip, nPip] | Internal thermal resistances [(m.K)/W] |
Modelica definition
function multipoleThermalResistances
"Thermal resistances from multipole solution"
extends Modelica.Icons.Function;
input Integer nPip "Number of pipes";
input Integer J "Number of multipoles";
input Modelica.SIunits.Position xPip[nPip] "x-Coordinates of pipes";
input Modelica.SIunits.Position yPip[nPip] "y-Coordinates of pipes";
input Modelica.SIunits.Radius rBor "Borehole radius";
input Modelica.SIunits.Radius rPip[nPip] "Outter radius of pipes";
input Modelica.SIunits.ThermalConductivity kFil "Thermal conductivity of grouting material";
input Modelica.SIunits.ThermalConductivity kSoi "Thermal conductivity of soil material";
input Real RFluPip[nPip](each unit="(m.K)/W") "Fluid to pipe wall thermal resistances";
input Modelica.SIunits.Temperature TBor=0 "Average borehole wall temperature";
output Real RDelta[nPip,nPip](each unit="(m.K)/W") "Delta-circuit thermal resistances";
output Real R[nPip,nPip](each unit="(m.K)/W") "Internal thermal resistances";
protected
Real QPip_flow[nPip](each unit="W/m") "Pipe heat transfer rates";
Modelica.SIunits.Temperature TFlu[nPip] "Fluid temperatures";
Real K[nPip,nPip](each unit="W/(m.K)") "Internal thermal conductances";
algorithm
for m in 1:nPip loop
for n in 1:nPip loop
if n == m then
QPip_flow[n] := 1;
else
QPip_flow[n] := 0;
end if;
end for;
TFlu :=
Buildings.Fluid.Geothermal.Borefields.BaseClasses.Boreholes.BaseClasses.Functions.multipoleFluidTemperature(
nPip,
J,
xPip,
yPip,
QPip_flow,
TBor,
rBor,
rPip,
kFil,
kSoi,
RFluPip);
for n in 1:nPip loop
R[n, m] := TFlu[n];
end for;
end for;
K := -Modelica.Math.Matrices.inv(R);
for m in 1:nPip loop
K[m, m] := -K[m, m];
for n in 1:nPip loop
if m <> n then
K[m, m] := K[m, m] - K[m, n];
end if;
end for;
end for;
for m in 1:nPip loop
for n in 1:nPip loop
RDelta[m, n] := 1./K[m, n];
end for;
end for;
end multipoleThermalResistances;
Buildings.Fluid.Geothermal.Borefields.BaseClasses.Boreholes.BaseClasses.Functions.partialInternalResistances
Partial model for borehole resistance calculation
Information
This partial function defines the common inputs to functions that calculate the borehole internal resistances.
Extends from Modelica.Icons.Function (Icon for functions).
Inputs
| Type | Name | Default | Description |
|---|---|---|---|
| Boolean | use_Rb | false | True if the value Rb should be used instead of calculated |
| Real | Rb | Rb(unit="(m.K)/W") | Borehole thermal resistance [(m.K)/W] |
| Height | hSeg | Height of the element [m] | |
| Radius | rBor | Radius of the borehole [m] | |
| Radius | rTub | Radius of the tube [m] | |
| Length | eTub | Thickness of the tubes [m] | |
| Length | sha | Shank spacing, defined as the distance between the center of a pipe and the center of the borehole [m] | |
| ThermalConductivity | kFil | Thermal conductivity of the grout [W/(m.K)] | |
| ThermalConductivity | kSoi | Thermal conductivity of the soi [W/(m.K)] | |
| ThermalConductivity | kTub | Thermal conductivity of the tube [W/(m.K)] | |
| ThermalConductivity | kMed | Thermal conductivity of the fluid [W/(m.K)] | |
| DynamicViscosity | muMed | Dynamic viscosity of the fluid [Pa.s] | |
| SpecificHeatCapacity | cpMed | Specific heat capacity of the fluid [J/(kg.K)] | |
| MassFlowRate | m_flow_nominal | Nominal mass flow rate [kg/s] | |
| Boolean | printDebug | false | Print resistances values in log for debug purposes. |
Outputs
| Type | Name | Description |
|---|---|---|
| Real | x | Capacity location |
Modelica definition
partial function partialInternalResistances
"Partial model for borehole resistance calculation"
extends Modelica.Icons.Function;
// Geometry of the borehole
input Boolean use_Rb = false
"True if the value Rb should be used instead of calculated";
input Real Rb(unit="(m.K)/W") "Borehole thermal resistance";
input Modelica.SIunits.Height hSeg "Height of the element";
input Modelica.SIunits.Radius rBor "Radius of the borehole";
// Geometry of the pipe
input Modelica.SIunits.Radius rTub "Radius of the tube";
input Modelica.SIunits.Length eTub "Thickness of the tubes";
input Modelica.SIunits.Length sha
"Shank spacing, defined as the distance between the center of a pipe and the center of the borehole";
// Thermal properties
input Modelica.SIunits.ThermalConductivity kFil
"Thermal conductivity of the grout";
input Modelica.SIunits.ThermalConductivity kSoi
"Thermal conductivity of the soi";
input Modelica.SIunits.ThermalConductivity kTub
"Thermal conductivity of the tube";
input Modelica.SIunits.ThermalConductivity kMed
"Thermal conductivity of the fluid";
input Modelica.SIunits.DynamicViscosity muMed
"Dynamic viscosity of the fluid";
input Modelica.SIunits.SpecificHeatCapacity cpMed
"Specific heat capacity of the fluid";
input Modelica.SIunits.MassFlowRate m_flow_nominal "Nominal mass flow rate";
input Boolean printDebug=false
"Print resistances values in log for debug purposes.";
// Outputs
output Real x "Capacity location";
protected
parameter Real pi = 3.141592653589793 "pi";
parameter Real rTub_in = rTub-eTub "Inner radius of tube";
Real RConv(unit="(m.K)/W")=
Buildings.Fluid.Geothermal.Borefields.BaseClasses.Boreholes.BaseClasses.Functions.convectionResistanceCircularPipe(
hSeg=hSeg,
rTub=rTub,
eTub=eTub,
kMed=kMed,
muMed=muMed,
cpMed=cpMed,
m_flow=m_flow_nominal,
m_flow_nominal=m_flow_nominal)*hSeg;
Boolean test=false "thermodynamic test for R and x value";
Real RCondPipe(unit="(m.K)/W") = Modelica.Math.log((rTub)/rTub_in)/(2*Modelica.Constants.pi*kTub)
"Thermal resistance of the pipe wall";
Real Rb_internal(unit="(m.K)/W")
"Resistance from the fluid in the pipe to the borehole wall";
Real Rb_multipole(unit="(m.K)/W")
"Theoretical Fluid-to-borehole-wall resistance evaluated from the multipole method";
Integer i=1 "Loop counter";
end partialInternalResistances;