Buildings.Fluid.HeatExchangers.Boreholes.BaseClasses

Information

This package contains base classes that are used to construct the models in Buildings.Fluid.HeatExchangers.Boreholes.

Package Content

Name	Description
BoreholeSegment	Vertical segment of a borehole
HexInternalElement	Internal part of a borehole
BoreholeResistance	Thermal resistance of the borehole
InterferenceResistance	Total interference resistance between the two pipes of a borehole
PartialBoreholeResistance	Partial model for thermal resistance of the borehole
SingleUTubeBoundaryCondition	Prescribed temperature at the outer boundary of a single U tube borehole
ExtendableArray	class used to create the external object: ExtendableArray
boreholeInternalResistance	Calculate the internal resistance of the borehole
boreholeThermalResistance	Calculate the total thermal resistance of the borehole from the fluid to the soil
exchangeValues	Store the value u at the element n in the external object, and return the value of the element i
factorial	Calculates the factorial value of the input
powerSeries	Power series used to compute far-field temperature
temperatureDrop	Calculate the temperature drop of the soil at the external boundary of the cylinder
Examples	Example models to test base classes

Buildings.Fluid.HeatExchangers.Boreholes.BaseClasses.BoreholeSegment

Information

Horizontal layer that is used to model a U-tube borehole heat exchanger. This model combines three models, each simulating a different aspect of a borehole heat exchanger.

The instance pipFil computes the heat transfer in the pipes and the filling material. This computation is done using the model Buildings.Fluid.Boreholes.BaseClasses.HexInternalElement.

The instance soi computes transient and steady state heat transfer in the soil using a vertical cylinder. The computation is done using the model Buildings.HeatTransfer.Conduction.SingleLayerCylinder.

The model TBouCon computes the far-field temperature boundary condition, i.e., the temperature at the outer surface of the above cylindrical heat transfer computation. The computation is done using the model Buildings.Fluid.Boreholes.BaseClasses.TemperatureBoundaryCondition.

Parameters

Type	Name	Default	Description
replaceable package Medium1		PartialMedium	Medium 1 in the component
replaceable package Medium2		PartialMedium	Medium 2 in the component
replaceable package Medium		PartialMedium	Medium in the component
Radius	rBor	0.1	Radius of the borehole [m]
Height	hSeg		Height of the element [m]
Length	xC	0.05	Shank spacing, defined as the distance between the center of a pipe and the center of the borehole [m]
Real	B0	17.44	Shape coefficient for grout resistance
Real	B1	-0.6052	Shape coefficient for grout resistance
Nominal condition
MassFlowRate	m1_flow_nominal	m_flow_nominal	Nominal mass flow rate [kg/s]
MassFlowRate	m2_flow_nominal	m_flow_nominal	Nominal mass flow rate [kg/s]
Pressure	dp_nominal		Pressure [Pa]
MassFlowRate	m_flow_nominal		Nominal mass flow rate [kg/s]
Initialization
MassFlowRate	m1_flow.start	0	Mass flow rate from port_a1 to port_b1 (m1_flow > 0 is design flow direction) [kg/s]
Pressure	dp1.start	0	Pressure difference between port_a1 and port_b1 [Pa]
MassFlowRate	m2_flow.start	0	Mass flow rate from port_a2 to port_b2 (m2_flow > 0 is design flow direction) [kg/s]
Pressure	dp2.start	0	Pressure difference between port_a2 and port_b2 [Pa]
Soil
Generic	matSoi	redeclare parameter Building...	Thermal properties of soil
Radius	rExt	3	Radius of the soil used for the external boundary condition [m]
Temperature	TExt_start	283.15	Initial far field temperature [K]
Integer	nSta	10	Number of state variables in the soil
Time	samplePeriod	604800	Sample period for the external boundary condition [s]
Filling material
Generic	matFil	redeclare parameter Building...	Thermal properties of the filling material
Temperature	TFil_start	283.15	Initial temperature of the filling material [K]
Tubes
Radius	rTub	0.02	Radius of the tubes [m]
ThermalConductivity	kTub	0.5	Thermal conductivity of the tubes [W/(m.K)]
Length	eTub	0.002	Thickness of the tubes [m]
Assumptions
Boolean	allowFlowReversal1	allowFlowReversal	= true to allow flow reversal in medium 1, false restricts to design direction (port_a -> port_b)
Boolean	allowFlowReversal2	allowFlowReversal	= true to allow flow reversal in medium 2, false restricts to design direction (port_a -> port_b)
Boolean	allowFlowReversal	true	= true to allow flow reversal, false restricts to design direction (port_a -> port_b)
Advanced
Initialization
SpecificEnthalpy	h_outflow_a1_start	Medium1.h_default	Start value for enthalpy flowing out of port a1 [J/kg]
SpecificEnthalpy	h_outflow_b1_start	Medium1.h_default	Start value for enthalpy flowing out of port b1 [J/kg]
SpecificEnthalpy	h_outflow_a2_start	Medium2.h_default	Start value for enthalpy flowing out of port a2 [J/kg]
SpecificEnthalpy	h_outflow_b2_start	Medium2.h_default	Start value for enthalpy flowing out of port b2 [J/kg]
MassFlowRate	m1_flow_small	m_flow_small	Small mass flow rate for regularization of zero flow [kg/s]
MassFlowRate	m2_flow_small	m_flow_small	Small mass flow rate for regularization of zero flow [kg/s]
Boolean	homotopyInitialization	true	= true, use homotopy method
MassFlowRate	m_flow_small	1E-4*abs(m_flow_nominal)	Small mass flow rate for regularization of zero flow [kg/s]
Diagnostics
Boolean	show_T	false	= true, if actual temperature at port is computed
Flow resistance
Boolean	computeFlowResistance	true	=true, compute flow resistance. Set to false to assume no friction
Boolean	from_dp	false	= true, use m_flow = f(dp) else dp = f(m_flow)
Boolean	linearizeFlowResistance	false	= true, use linear relation between m_flow and dp for any flow rate
Real	deltaM	0.1	Fraction of nominal flow rate where flow transitions to laminar
Dynamics
Equations
Dynamics	energyDynamics	Modelica.Fluid.Types.Dynamic...	Formulation of energy balance
Dynamics	massDynamics	energyDynamics	Formulation of mass balance
Initialization
AbsolutePressure	p_start	Medium.p_default	Start value of pressure [Pa]
Temperature	T_start	TFil_start	Start value of temperature [K]
MassFraction	X_start[Medium.nX]	Medium.X_default	Start value of mass fractions m_i/m [kg/kg]
ExtraProperty	C_start[Medium.nC]	fill(0, Medium.nC)	Start value of trace substances
ExtraProperty	C_nominal[Medium.nC]	fill(1E-2, Medium.nC)	Nominal value of trace substances. (Set to typical order of magnitude.)

Connectors

Type	Name	Description
replaceable package Medium1		Medium 1 in the component
replaceable package Medium2		Medium 2 in the component
FluidPort_a	port_a1	Fluid connector a1 (positive design flow direction is from port_a1 to port_b1)
FluidPort_b	port_b1	Fluid connector b1 (positive design flow direction is from port_a1 to port_b1)
FluidPort_a	port_a2	Fluid connector a2 (positive design flow direction is from port_a2 to port_b2)
FluidPort_b	port_b2	Fluid connector b2 (positive design flow direction is from port_a2 to port_b2)
replaceable package Medium		Medium in the component

Modelica definition

model BoreholeSegment "Vertical segment of a borehole"
  extends Buildings.Fluid.Interfaces.PartialFourPortInterface(
     redeclare final package Medium1 = Medium,
     redeclare final package Medium2 = Medium,
     final m1_flow_nominal = m_flow_nominal,
     final m2_flow_nominal = m_flow_nominal,
     final m1_flow_small =   m_flow_small,
     final m2_flow_small =   m_flow_small,
     final allowFlowReversal1=allowFlowReversal,
     final allowFlowReversal2=allowFlowReversal);
  extends Buildings.Fluid.Interfaces.TwoPortFlowResistanceParameters;
  extends Buildings.Fluid.Interfaces.LumpedVolumeDeclarations(T_start=TFil_start);
  replaceable package Medium = Modelica.Media.Interfaces.PartialMedium 
    "Medium in the component";
  replaceable parameter Buildings.HeatTransfer.Data.Soil.Generic matSoi 
    "Thermal properties of soil";
  replaceable parameter Buildings.HeatTransfer.Data.BoreholeFillings.Generic matFil 
    "Thermal properties of the filling material";

  parameter Modelica.SIunits.MassFlowRate m_flow_nominal 
    "Nominal mass flow rate";
  parameter Modelica.SIunits.MassFlowRate m_flow_small(min=0) = 1E-4*abs(m_flow_nominal) 
    "Small mass flow rate for regularization of zero flow";
  parameter Boolean homotopyInitialization = true "= true, use homotopy method";

  parameter Modelica.SIunits.Radius rTub=0.02 "Radius of the tubes";
  parameter Modelica.SIunits.ThermalConductivity kTub=0.5 
    "Thermal conductivity of the tubes";
  parameter Modelica.SIunits.Length eTub=0.002 "Thickness of the tubes";
  parameter Modelica.SIunits.Temperature TFil_start=283.15 
    "Initial temperature of the filling material";
  parameter Modelica.SIunits.Radius rExt=3 
    "Radius of the soil used for the external boundary condition";
  parameter Modelica.SIunits.Temperature TExt_start=283.15 
    "Initial far field temperature";
  parameter Integer nSta(min=1) = 10 "Number of state variables in the soil";
  parameter Modelica.SIunits.Time samplePeriod=604800 
    "Sample period for the external boundary condition";
  parameter Modelica.SIunits.Radius rBor=0.1 "Radius of the borehole";
  parameter Modelica.SIunits.Height hSeg "Height of the element";
  parameter Modelica.SIunits.Length xC=0.05 
    "Shank spacing, defined as the distance between the center of a pipe and the center of the borehole";
  parameter Real B0=17.44 "Shape coefficient for grout resistance";
  parameter Real B1=-0.6052 "Shape coefficient for grout resistance";

 parameter Boolean allowFlowReversal = true 
    "= true to allow flow reversal, false restricts to design direction (port_a -> port_b)";

 Buildings.Fluid.HeatExchangers.Boreholes.BaseClasses.HexInternalElement pipFil(
    redeclare final package Medium = Medium,
    final matFil=matFil,
    final hSeg=hSeg,
    final rTub=rTub,
    final eTub=eTub,
    final kTub=kTub,
    final kSoi=matSoi.k,
    final xC=xC,
    final rBor=rBor,
    final TFil_start=TFil_start,
    final B0=B0,
    final B1=B1,
    final m1_flow_nominal=m_flow_nominal,
    final m2_flow_nominal=m_flow_nominal,
    final dp1_nominal=dp_nominal,
    final dp2_nominal=0,
    final from_dp1=from_dp,
    final from_dp2=from_dp,
    final linearizeFlowResistance1=linearizeFlowResistance,
    final linearizeFlowResistance2=linearizeFlowResistance,
    final deltaM1=deltaM,
    final deltaM2=deltaM,
    final m1_flow_small=m_flow_small,
    final m2_flow_small=m_flow_small,
    final allowFlowReversal1=allowFlowReversal,
    final allowFlowReversal2=allowFlowReversal,
    final homotopyInitialization=homotopyInitialization,
    final energyDynamics=energyDynamics,
    final massDynamics=massDynamics,
    final p1_start=p_start,
    T1_start=T_start,
    X1_start=X_start,
    C1_start=C_start,
    C1_nominal=C_nominal,
    final p2_start=p_start,
    T2_start=T_start,
    X2_start=X_start,
    C2_start=C_start,
    C2_nominal=C_nominal) 
    "Internal part of the borehole including the pipes and the filling material";
  Buildings.HeatTransfer.Conduction.SingleLayerCylinder soi(
    final material=matSoi,
    final h=hSeg,
    final nSta=nSta,
    final r_a=rBor,
    final r_b=rExt,
    final steadyStateInitial=false,
    final TInt_start=TFil_start,
    final TExt_start=TExt_start) "Heat conduction in the soil";
  Buildings.Fluid.HeatExchangers.Boreholes.BaseClasses.SingleUTubeBoundaryCondition
    TBouCon(
    final matSoi=matSoi,
    final rExt=rExt,
    final hSeg=hSeg,
    final TExt_start=TExt_start,
    final samplePeriod=samplePeriod) 
    "Thermal boundary condition for the far-field";
protected 
  Modelica.Thermal.HeatTransfer.Sensors.HeatFlowSensor heaFlo;
equation 
  connect(pipFil.port_b1, port_b1);
  connect(pipFil.port_a2, port_a2);
  connect(pipFil.port_b2, port_b2);
  connect(pipFil.port, heaFlo.port_a);
  connect(heaFlo.port_b, soi.port_a);
  connect(soi.port_b, TBouCon.port);
  connect(port_a1, pipFil.port_a1);
  connect(heaFlo.Q_flow, TBouCon.Q_flow);
end BoreholeSegment;

Buildings.Fluid.HeatExchangers.Boreholes.BaseClasses.HexInternalElement

Information

Model for the heat transfer between the fluid and within the borehole filling. This model computes the dynamic response of the fluid in the tubes, and the heat transfer between the fluid and the borehole filling, and the heat storage within the fluid and the borehole filling.

The heat conduction in the filling material is modeled using three three resistances model that are arranged in a triangular configuration. Two of these resistances represent the heat conduction from the fluids to the external radius of the borehole, and the other resistance represents the thermal interference between the two pipes.

The resistance between the fluid and the borehole wall are the sum of the convective resistance inside the tubes, the conductive resistance of the tube wall and the conductive resistance of the filling material. They are obtained using

G_Con = 2 π h_seg r_tub h_in ,

G_tub = 4 π k_tub h_seg ⁄ ln( ( r_tub+e_tub ) ⁄ r_tub ),

G_fil= k_fil h_seg β₀ ( r_Bor ⁄ r_tub ) ^β₁ ,

where h_seg is the height of the tube, h_in is the convection coefficient, k_tub is the thermal conductivity of the tube, e_tub is the thickness of the tube, and β₀, β₁ are the resistance shape factor coefficients (Paul, 1996). Paul's shape factors are based on experimental and finite element analysis of typical borehole. The default values used for these coefficients are β₀= 20.100 and β₁=-0.94467. Values listed by Paul are given in the table below.

Implementation

The resistances between the fluid and the borehole wall are computed in Buildings.Fluid.HeatExchangers.Boreholes.BaseClasses.BoreholeResistance. The resistance for the interference of the pipes is computed in Buildings.Fluid.HeatExchangers.Boreholes.BaseClasses.InterferenceResistance.

Parameters

Type	Name	Default	Description
replaceable package Medium1		PartialMedium	Medium 1 in the component
replaceable package Medium2		PartialMedium	Medium 2 in the component
replaceable package Medium		Modelica.Media.Interfaces.Pa...	Medium in the component
ThermalConductivity	kSoi		Thermal conductivity of the soil used for the calculation of the internal interference resistance [W/(m.K)]
Radius	rBor		Radius of the borehole [m]
Height	hSeg		Height of the element [m]
Real	B0	17.44268	Shape coefficient for grout resistance
Real	B1	-0.60515	Shape coefficient for grout resistance
Length	xC	0.05	Shank spacing definied as half the center-to-center distance between the two pipes [m]
Nominal condition
MassFlowRate	m1_flow_nominal		Nominal mass flow rate [kg/s]
MassFlowRate	m2_flow_nominal		Nominal mass flow rate [kg/s]
Pressure	dp1_nominal		Pressure [Pa]
Pressure	dp2_nominal		Pressure [Pa]
Initialization
MassFlowRate	m1_flow.start	0	Mass flow rate from port_a1 to port_b1 (m1_flow > 0 is design flow direction) [kg/s]
Pressure	dp1.start	0	Pressure difference between port_a1 and port_b1 [Pa]
MassFlowRate	m2_flow.start	0	Mass flow rate from port_a2 to port_b2 (m2_flow > 0 is design flow direction) [kg/s]
Pressure	dp2.start	0	Pressure difference between port_a2 and port_b2 [Pa]
Filling material
Generic	matFil	redeclare parameter Building...	Thermal properties of the filling material
Temperature	TFil_start	283.15	Initial temperature of the filling material [K]
Pipes
Radius	rTub	0.02	Radius of the tubes [m]
ThermalConductivity	kTub	0.5	Thermal conductivity of the tubes [W/(m.K)]
Length	eTub	0.002	Thickness of the tubes [m]
Assumptions
Boolean	allowFlowReversal1	system.allowFlowReversal	= true to allow flow reversal in medium 1, false restricts to design direction (port_a -> port_b)
Boolean	allowFlowReversal2	system.allowFlowReversal	= true to allow flow reversal in medium 2, false restricts to design direction (port_a -> port_b)
Advanced
MassFlowRate	m1_flow_small	1E-4*abs(m1_flow_nominal)	Small mass flow rate for regularization of zero flow [kg/s]
MassFlowRate	m2_flow_small	1E-4*abs(m2_flow_nominal)	Small mass flow rate for regularization of zero flow [kg/s]
Boolean	homotopyInitialization	true	= true, use homotopy method
Diagnostics
Boolean	show_T	false	= true, if actual temperature at port is computed
Flow resistance
Medium 1
Boolean	from_dp1	false	= true, use m_flow = f(dp) else dp = f(m_flow)
Boolean	linearizeFlowResistance1	false	= true, use linear relation between m_flow and dp for any flow rate
Real	deltaM1	0.1	Fraction of nominal flow rate where flow transitions to laminar
Medium 2
Boolean	from_dp2	false	= true, use m_flow = f(dp) else dp = f(m_flow)
Boolean	linearizeFlowResistance2	false	= true, use linear relation between m_flow and dp for any flow rate
Real	deltaM2	0.1	Fraction of nominal flow rate where flow transitions to laminar
Dynamics
Nominal condition
Time	tau1	Modelica.Constants.pi*rTub^2...	Time constant at nominal flow [s]
Time	tau2	Modelica.Constants.pi*rTub^2...	Time constant at nominal flow [s]
Equations
Dynamics	energyDynamics	Modelica.Fluid.Types.Dynamic...	Formulation of energy balance
Dynamics	massDynamics	energyDynamics	Formulation of mass balance
Initialization
Medium 1
AbsolutePressure	p1_start	Medium1.p_default	Start value of pressure [Pa]
Temperature	T1_start	TFil_start	Start value of temperature [K]
MassFraction	X1_start[Medium1.nX]	Medium1.X_default	Start value of mass fractions m_i/m [kg/kg]
ExtraProperty	C1_start[Medium1.nC]	fill(0, Medium1.nC)	Start value of trace substances
ExtraProperty	C1_nominal[Medium1.nC]	fill(1E-2, Medium1.nC)	Nominal value of trace substances. (Set to typical order of magnitude.)
Medium 2
AbsolutePressure	p2_start	Medium2.p_default	Start value of pressure [Pa]
Temperature	T2_start	TFil_start	Start value of temperature [K]
MassFraction	X2_start[Medium2.nX]	Medium2.X_default	Start value of mass fractions m_i/m [kg/kg]
ExtraProperty	C2_start[Medium2.nC]	fill(0, Medium2.nC)	Start value of trace substances
ExtraProperty	C2_nominal[Medium2.nC]	fill(1E-2, Medium2.nC)	Nominal value of trace substances. (Set to typical order of magnitude.)

Connectors

Type	Name	Description
replaceable package Medium1		Medium 1 in the component
replaceable package Medium2		Medium 2 in the component
FluidPort_a	port_a1	Fluid connector a1 (positive design flow direction is from port_a1 to port_b1)
FluidPort_b	port_b1	Fluid connector b1 (positive design flow direction is from port_a1 to port_b1)
FluidPort_a	port_a2	Fluid connector a2 (positive design flow direction is from port_a2 to port_b2)
FluidPort_b	port_b2	Fluid connector b2 (positive design flow direction is from port_a2 to port_b2)
replaceable package Medium		Medium in the component
HeatPort_a	port	Heat port that connects to filling material

Modelica definition

model HexInternalElement "Internal part of a borehole"
  extends Buildings.Fluid.Interfaces.FourPortHeatMassExchanger(
    redeclare final package Medium1 = Medium,
    redeclare final package Medium2 = Medium,
    T1_start=TFil_start,
    T2_start=TFil_start,
    final tau1=Modelica.Constants.pi*rTub^2*hSeg*rho1_nominal/m1_flow_nominal,
    final tau2=Modelica.Constants.pi*rTub^2*hSeg*rho2_nominal/m2_flow_nominal,
    final show_T=false,
    vol1(final energyDynamics=energyDynamics,
         final massDynamics=massDynamics,
         final prescribedHeatFlowRate=false,
         final homotopyInitialization=homotopyInitialization,
         final allowFlowReversal=allowFlowReversal1,
         final V=m2_flow_nominal*tau2/rho2_nominal,
         final m_flow_small=m1_flow_small),
    final vol2(final energyDynamics=energyDynamics,
         final massDynamics=massDynamics,
         final prescribedHeatFlowRate=false,
         final homotopyInitialization=homotopyInitialization,
         final V=m1_flow_nominal*tau1/rho1_nominal,
         final m_flow_small=m2_flow_small));

  replaceable package Medium = Modelica.Media.Interfaces.PartialMedium 
    "Medium in the component";

  replaceable parameter Buildings.HeatTransfer.Data.BoreholeFillings.Generic matFil 
    "Thermal properties of the filling material";

  parameter Modelica.SIunits.Radius rTub=0.02 "Radius of the tubes";
  parameter Modelica.SIunits.ThermalConductivity kTub=0.5 
    "Thermal conductivity of the tubes";
  parameter Modelica.SIunits.Length eTub=0.002 "Thickness of the tubes";
  parameter Modelica.SIunits.ThermalConductivity kSoi 
    "Thermal conductivity of the soil used for the calculation of the internal interference resistance";

  parameter Modelica.SIunits.Temperature TFil_start=283.15 
    "Initial temperature of the filling material";

  parameter Modelica.SIunits.Radius rBor "Radius of the borehole";
  parameter Modelica.SIunits.Height hSeg "Height of the element";

  parameter Real B0=17.44268 "Shape coefficient for grout resistance";
  parameter Real B1=-0.60515 "Shape coefficient for grout resistance";

  parameter Modelica.SIunits.Length xC=0.05 
    "Shank spacing definied as half the center-to-center distance between the two pipes";

  Modelica.Thermal.HeatTransfer.Interfaces.HeatPort_a port 
    "Heat port that connects to filling material";
protected 
  final parameter Modelica.SIunits.SpecificHeatCapacity cpFil=matFil.c 
    "Specific heat capacity of the filling material";
  final parameter Modelica.SIunits.ThermalConductivity kFil=matFil.k 
    "Thermal conductivity of the filling material";
  final parameter Modelica.SIunits.Density dFil=matFil.d 
    "Density of the filling material";
  parameter Modelica.SIunits.HeatCapacity Co_fil=0.5*dFil*cpFil*hSeg*Modelica.Constants.pi
      *(rBor^2 - 2*(rTub + eTub)^2) "Heat capacity of the filling material";
  Modelica.Thermal.HeatTransfer.Components.HeatCapacitor cap2(C=Co_fil, T(start=
         TFil_start)) "Heat capacity of the filling material";
  Modelica.Thermal.HeatTransfer.Components.HeatCapacitor cap1(C=Co_fil,T(start=
          TFil_start)) "Heat capacity of the filling material";
  BoreholeResistance con1(
    redeclare final package Medium = Medium,
    final rBor=rBor,
    final rTub=rTub,
    final eTub=eTub,
    final kTub=kTub,
    final kFil=kFil,
    final hSeg=hSeg,
    final B0=B0,
    final B1=B1,
    final m_flow_nominal=m1_flow_nominal,
    homotopyInitialization=homotopyInitialization) 
    "Thermal resistance from the fluid to the wall of the borehole";
  BoreholeResistance con2(
    redeclare final package Medium = Medium,
    final rBor=rBor,
    final rTub=rTub,
    final eTub=eTub,
    final kTub=kTub,
    final kFil=kFil,
    final hSeg=hSeg,
    final B0=B0,
    final B1=B1,
    final m_flow_nominal=m1_flow_nominal,
    homotopyInitialization=homotopyInitialization) 
    "Thermal resistance from the fluid to the wall of the borehole";

  InterferenceResistance con12(
    redeclare final package Medium = Medium,
    final rBor=rBor,
    final rTub=rTub,
    final eTub=eTub,
    final kTub=kTub,
    final kFil=kFil,
    final kSoi=kSoi,
    final hSeg=hSeg,
    final xC=xC,
    final B0=B0,
    final B1=B1,
    final m_flow_nominal=m1_flow_nominal,
    homotopyInitialization=homotopyInitialization) 
    "Interference resistance between the two pipes";

  Modelica.Blocks.Sources.RealExpression m_flow(y=port_a1.m_flow) 
    "Mass flow rate";
equation 
  connect(vol1.heatPort, cap1.port);
  connect(vol2.heatPort, cap2.port);
  connect(cap1.port, con12.port_a);
  connect(cap1.port, con1.port_a);
  connect(con12.port_b, cap2.port);
  connect(con2.port_a, cap2.port);
  connect(con1.port_b, port);
  connect(con2.port_b, port);
  connect(m_flow.y, con1.m_flow);
  connect(m_flow.y, con2.m_flow);
  connect(m_flow.y, con12.m_flow);
end HexInternalElement;

Buildings.Fluid.HeatExchangers.Boreholes.BaseClasses.BoreholeResistance

Information

This model computes the total thermal resistance from the fluid to the borehole wall. This thermal resistance is then used for a three-resistance model. It assumes that the pipe configuration is symmetric. The resistance is obtain using

R = 2 R_b,

where R_b is the total borehole resistance as defined by Hellstrom (1991).

The resistance R_b includes three different thermal resistances:

1. a convective resistance from the brine to the pipe wall,
2. a thermal resistance of the pipe wall, and
3. the thermal resistance of the filling material.

Implementation

The calculation of R_b is done using the function Buildings.Fluid.Boreholes.BaseClasses.boreholeThermalResistance.

References

Hellstrom, G (1991).
Thermal Performance of Borehole Heat Exchangers.
Department of Mathematical Physics, Lund Institute of Technology.

Parameters

Type	Name	Default	Description
replaceable package Medium		PartialMedium	Medium in the component
Radius	rBor		Radius of the borehole [m]
ThermalConductivity	kFil		[W/(m.K)]
Height	hSeg		Height of the segment [m]
Real	B0	17.44268	Shape coefficient for grout resistance
Real	B1	-0.60515	Shape coefficient for grout resistance
Pipes
Radius	rTub		Radius of the tubes [m]
ThermalConductivity	kTub	4	Thermal conductivity of the tubes [W/(m.K)]
Length	eTub	0.002	Thickness of the tubes [m]
Nominal condition
MassFlowRate	m_flow_nominal		Nominal mass flow rate [kg/s]
Advanced
Boolean	homotopyInitialization	true	= true, use homotopy method

Connectors

Type	Name	Description
input RealInput	m_flow	Mass flow rate [kg/s]
HeatPort_a	port_a
HeatPort_b	port_b

Modelica definition

model BoreholeResistance "Thermal resistance of the borehole"
  extends Buildings.Fluid.HeatExchangers.Boreholes.BaseClasses.PartialBoreholeResistance;

equation 
  if homotopyInitialization then
    Rb = homotopy(actual=Buildings.Fluid.HeatExchangers.Boreholes.BaseClasses.boreholeThermalResistance(
          hSeg=hSeg, rBor=rBor, rTub=rTub, eTub=eTub, kTub=kTub, kFil=kFil,
          kMed=kMed, mueMed=mueMed, cpFluid=cpFluid, m_flow=m_flow, m_flow_nominal=m_flow_nominal,
          B0=B0, B1=B1),
                  simplified=  Buildings.Fluid.HeatExchangers.Boreholes.BaseClasses.boreholeThermalResistance(
          hSeg=hSeg, rBor=rBor, rTub=rTub, eTub=eTub, kTub=kTub, kFil=kFil,
          kMed=kMed, mueMed=mueMed, cpFluid=cpFluid, m_flow=m_flow_nominal, m_flow_nominal=m_flow_nominal,
          B0=B0, B1=B1));
  else
    Rb = Buildings.Fluid.HeatExchangers.Boreholes.BaseClasses.boreholeThermalResistance(
          hSeg=hSeg, rBor=rBor, rTub=rTub, eTub=eTub, kTub=kTub, kFil=kFil,
          kMed=kMed, mueMed=mueMed, cpFluid=cpFluid, m_flow=m_flow, m_flow_nominal=m_flow_nominal,
          B0=B0, B1=B1);
  end if;
  G = 1/(2*Rb);
end BoreholeResistance;

Buildings.Fluid.HeatExchangers.Boreholes.BaseClasses.InterferenceResistance

Information

This model computes the interference resistance between the two pipes of a U-tube borehole. It computes the resistance using the internal resistance R_a and the borehole total resistance R_b.

The coupling resistance is obtained using

R_i = 4 R_b R_a ⁄ (4 R_b- R_a).

Implementation

The two resistances are calculated by the functions Buildings.Fluid.HeatExchangers.Boreholes.BaseClasses.boreholeThermalResistance for R_b and Buildings.Fluid.HeatExchangers.Boreholes.BaseClasses.boreholeInternalResistance for R_a.

Parameters

Type	Name	Default	Description
replaceable package Medium		PartialMedium	Medium in the component
Radius	rBor		Radius of the borehole [m]
ThermalConductivity	kFil		[W/(m.K)]
Height	hSeg		Height of the segment [m]
Real	B0	17.44268	Shape coefficient for grout resistance
Real	B1	-0.60515	Shape coefficient for grout resistance
Length	xC		Shank spacing definied as half the center-to-center distance between the two pipes [m]
ThermalConductivity	kSoi		Thermal conductivity of the soil used for the calculation of the internal interference resistance [W/(m.K)]
Pipes
Radius	rTub		Radius of the tubes [m]
ThermalConductivity	kTub	4	Thermal conductivity of the tubes [W/(m.K)]
Length	eTub	0.002	Thickness of the tubes [m]
Nominal condition
MassFlowRate	m_flow_nominal		Nominal mass flow rate [kg/s]
Advanced
Boolean	homotopyInitialization	true	= true, use homotopy method

Connectors

Type	Name	Description
input RealInput	m_flow	Mass flow rate [kg/s]
HeatPort_a	port_a
HeatPort_b	port_b

Modelica definition

model InterferenceResistance 
  "Total interference resistance between the two pipes of a borehole"
  extends Buildings.Fluid.HeatExchangers.Boreholes.BaseClasses.PartialBoreholeResistance;

  parameter Modelica.SIunits.Length xC 
    "Shank spacing definied as half the center-to-center distance between the two pipes";

  parameter Modelica.SIunits.ThermalConductivity kSoi 
    "Thermal conductivity of the soil used for the calculation of the internal interference resistance";

protected 
  parameter Modelica.SIunits.ThermalResistance Ra=boreholeInternalResistance(
      hSeg=hSeg,
      rBor=rBor,
      rTub=rTub,
      eTub=eTub,
      xC=xC,
      kFil=kFil,
      kSoi=kSoi) "Borehole internal resistance";

equation 
  if homotopyInitialization then
    Rb = homotopy(actual=boreholeThermalResistance(
            hSeg=hSeg, rBor=rBor, rTub=rTub, eTub=eTub, kTub=kTub,
            kFil=kFil, kMed=kMed, mueMed=mueMed, cpFluid=cpFluid, m_flow=m_flow,
            m_flow_nominal=m_flow_nominal, B0=B0, B1=B1),
         simplified=boreholeThermalResistance(
            hSeg=hSeg, rBor=rBor, rTub=rTub, eTub=eTub, kTub=kTub,
            kFil=kFil, kMed=kMed, mueMed=mueMed, cpFluid=cpFluid, m_flow=m_flow_nominal,
            m_flow_nominal=m_flow_nominal, B0=B0, B1=B1));
  else
   Rb = boreholeThermalResistance(
    hSeg=hSeg, rBor=rBor, rTub=rTub, eTub=eTub, kTub=kTub,
    kFil=kFil, kMed=kMed, mueMed=mueMed, cpFluid=cpFluid, m_flow=m_flow,
    m_flow_nominal=m_flow_nominal, B0=B0, B1=B1);
  end if;
  G = (4*Rb - Ra)/(4*Rb*Ra);
end InterferenceResistance;

Buildings.Fluid.HeatExchangers.Boreholes.BaseClasses.PartialBoreholeResistance

Information

Partial model for the thermal resistance of the borehole.

Parameters

Type	Name	Default	Description
replaceable package Medium		Modelica.Media.Interfaces.Pa...	Medium in the component
Radius	rBor		Radius of the borehole [m]
ThermalConductivity	kFil		[W/(m.K)]
Height	hSeg		Height of the segment [m]
Real	B0	17.44268	Shape coefficient for grout resistance
Real	B1	-0.60515	Shape coefficient for grout resistance
Pipes
Radius	rTub		Radius of the tubes [m]
ThermalConductivity	kTub	4	Thermal conductivity of the tubes [W/(m.K)]
Length	eTub	0.002	Thickness of the tubes [m]
Nominal condition
MassFlowRate	m_flow_nominal		Nominal mass flow rate [kg/s]
Advanced
Boolean	homotopyInitialization	true	= true, use homotopy method

Connectors

Type	Name	Description
replaceable package Medium		Medium in the component
input RealInput	m_flow	Mass flow rate [kg/s]
HeatPort_a	port_a
HeatPort_b	port_b

Modelica definition

partial model PartialBoreholeResistance 
  "Partial model for thermal resistance of the borehole"
  replaceable package Medium = Modelica.Media.Interfaces.PartialMedium 
    "Medium in the component";
  parameter Modelica.SIunits.Radius rTub "Radius of the tubes";
  parameter Modelica.SIunits.ThermalConductivity kTub=4 
    "Thermal conductivity of the tubes";
  parameter Modelica.SIunits.Length eTub=0.002 "Thickness of the tubes";
  parameter Modelica.SIunits.Radius rBor "Radius of the borehole";
  parameter Modelica.SIunits.ThermalConductivity kFil;
  parameter Modelica.SIunits.Height hSeg "Height of the segment";
  parameter Real B0=17.44268 "Shape coefficient for grout resistance";
  parameter Real B1=-0.60515 "Shape coefficient for grout resistance";
  parameter Modelica.SIunits.MassFlowRate m_flow_nominal 
    "Nominal mass flow rate";

  parameter Boolean homotopyInitialization = true "= true, use homotopy method";

  Modelica.Blocks.Interfaces.RealInput m_flow(unit="kg/s") "Mass flow rate";

  Modelica.SIunits.ThermalConductance G "Thermal conductance";
  Modelica.SIunits.HeatFlowRate Q_flow "Heat flow rate from port_a to port_b";
  Modelica.SIunits.TemperatureDifference dT "= port_a.T - port_b.T";
  Modelica.Thermal.HeatTransfer.Interfaces.HeatPort_a port_a;
  Modelica.Thermal.HeatTransfer.Interfaces.HeatPort_b port_b;

protected 
  parameter Modelica.SIunits.SpecificHeatCapacity cpFluid=
      Medium.specificHeatCapacityCp(Medium.setState_pTX(
      Medium.p_default,
      Medium.T_default,
      Medium.X_default)) "Specific heat capacity of the fluid";
  parameter Modelica.SIunits.ThermalConductivity kMed=
      Medium.thermalConductivity(Medium.setState_pTX(
      Medium.p_default,
      Medium.T_default,
      Medium.X_default)) "Thermal conductivity of the fluid";
  parameter Modelica.SIunits.DynamicViscosity mueMed=Medium.dynamicViscosity(
      Medium.setState_pTX(
      Medium.p_default,
      Medium.T_default,
      Medium.X_default)) "Dynamic viscosity of the fluid";
  Modelica.SIunits.ThermalResistance Rb "Borehole thermal resistance";
equation 
  dT = port_a.T - port_b.T;
  port_a.Q_flow = Q_flow;
  port_b.Q_flow = -Q_flow;
  Q_flow = G*dT;
end PartialBoreholeResistance;

Buildings.Fluid.HeatExchangers.Boreholes.BaseClasses.SingleUTubeBoundaryCondition

Information

This model computes the temperature boundary condition at the outer boundary of the borehole. It takes as an input the heat flow rate at the center of the borehole. This heat flow rate is averaged over the sample period. At each sampling interval, typically every one week, a new temperature boundary conditions is computed using the analytical solution to a line source heat transfer problem.

Implementation

The computation of the temperature change of the boundary is computed using the function Buildings.Fluid.HeatExchangers.Boreholes.BaseClasses.temperatureDrop.

Parameters

Type	Name	Default	Description
Generic	matSoi	redeclare parameter Building...	Thermal properties of the soil
Radius	rExt	3	Distance from the brine where the calculation is performed [m]
Height	hSeg	10	Height of the segment [m]
Temperature	TExt_start	283.15	Initial external temperature [K]
Time	samplePeriod	604800	Period between two samples [s]

Connectors

Type	Name	Description
input RealInput	Q_flow	Heat flow rate at the center of the borehole, positive if heat is added to soil [W]
HeatPort_b	port	Heat port

Modelica definition

model SingleUTubeBoundaryCondition 
  "Prescribed temperature at the outer boundary of a single U tube borehole"
  replaceable parameter Buildings.HeatTransfer.Data.Soil.Generic matSoi 
    "Thermal properties of the soil";
  parameter Modelica.SIunits.Radius rExt=3 
    "Distance from the brine where the calculation is performed";
  parameter Modelica.SIunits.Height hSeg=10 "Height of the segment";
  parameter Modelica.SIunits.Temperature TExt_start=283.15 
    "Initial external temperature";
  parameter Modelica.SIunits.Time samplePeriod=604800 
    "Period between two samples";
  ExtendableArray table=ExtendableArray() 
    "Extentable array, used to store history of rate of heat flows";
  Modelica.SIunits.HeatFlowRate QAve_flow 
    "Average heat flux over a time period";
  Modelica.Blocks.Interfaces.RealInput Q_flow(unit="W") 
    "Heat flow rate at the center of the borehole, positive if heat is added to soil";
  Modelica.Thermal.HeatTransfer.Interfaces.HeatPort_b port "Heat port";
protected 
  final parameter Modelica.SIunits.SpecificHeatCapacity c= matSoi.c 
    "Specific heat capacity of the soil";
  final parameter Modelica.SIunits.ThermalConductivity k= matSoi.k 
    "Thermal conductivity of the soil";
  final parameter Modelica.SIunits.Density d = matSoi.d "Density of the soil";
  Modelica.SIunits.Energy UOld "Internal energy at the previous period";
  Modelica.SIunits.Energy U 
    "Current internal energy, defined as U=0 for t=tStart";
  final parameter Modelica.SIunits.Time startTime(fixed=false) 
    "Start time of the simulation";
  Integer iSam(min=1) 
    "Counter for how many time the model was sampled. Defined as iSam=1 when called at t=0";
initial algorithm 
  U         := 0;
  UOld      := 0;
  startTime := time;
  iSam      := 1;
equation 
  der(U) = Q_flow;
algorithm 
  when initial() or sample(startTime,samplePeriod) then
    QAve_flow := (U-UOld)/samplePeriod;
    UOld      := U;
    port.T    := TExt_start + Buildings.Fluid.HeatExchangers.Boreholes.BaseClasses.temperatureDrop(
                                 table=table, iSam=iSam,
                                 Q_flow=QAve_flow, samplePeriod=samplePeriod,
                                 rExt=rExt, hSeg=hSeg,
                                 k=k, d=d, c=c);
    iSam := iSam+1;
  end when;

end SingleUTubeBoundaryCondition;

Buildings.Fluid.HeatExchangers.Boreholes.BaseClasses.ExtendableArray

Information

Class derived from ExternalObject having two local external function definition, named destructor and constructor respectively.

These functions create and release an external object that allows the the storage of real parameters in an array of extendable dimension.

Modelica definition

class ExtendableArray 
  "class used to create the external object: ExtendableArray"
extends ExternalObject;
    function constructor 
    "Construct an extendable array that can be used to store double values"
    output ExtendableArray table;
    external "C" table = initArray();
    end constructor;

  function destructor 
    "Release storage of table and close the external object"
    input ExtendableArray  table;
    external "C" freeArray(table);
  end destructor;
end ExtendableArray;

Buildings.Fluid.HeatExchangers.Boreholes.BaseClasses.boreholeInternalResistance

Information

This model computes the internal resistance R of a single U-tube borehole, as defined by Hellstrom (1991), as

k = ( 2 P₁ ( P₂² + 1 ) ^W ) / ( P₂ ( P₂² - 1 )^W ),

R = ln(k) / (π k_fil h_seg) ,

References

Inputs

Type	Name	Default	Description
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	xC		Shank spacing, defined as half the center-to-center distance between the two pipes [m]
ThermalConductivity	kFil		Thermal conductivity of the filling material [W/(m.K)]
ThermalConductivity	kSoi		Thermal conductivity of the soil [W/(m.K)]

Outputs

Type	Name	Description
ThermalResistance	R	Internal resistance of the borehole [K/W]

Modelica definition

function boreholeInternalResistance 
  "Calculate the internal resistance of the borehole"
  input Modelica.SIunits.Height hSeg "Height of the element";
  input Modelica.SIunits.Radius rBor "Radius of the borehole";
  input Modelica.SIunits.Radius rTub "Radius of the tube";
  input Modelica.SIunits.Length eTub "Thickness of the tubes";
  input Modelica.SIunits.Length xC 
    "Shank spacing, defined as half the center-to-center distance between the two pipes";
  input Modelica.SIunits.ThermalConductivity kFil 
    "Thermal conductivity of the filling material";
  input Modelica.SIunits.ThermalConductivity kSoi 
    "Thermal conductivity of the soil";
  output Modelica.SIunits.ThermalResistance R 
    "Internal resistance of the borehole";
protected 
  final parameter Real P1=rBor/(rTub + eTub) 
    "Non-dimensional geometrical parameter ";
  final parameter Real P2=rBor/xC "Non-dimensional geometrical parameter ";
  final parameter Real P3=(kFil - kSoi)/(kFil + kSoi) 
    "Non-dimensional parameter for filling and soil conductivity";
algorithm 
  assert(xC > (rTub + eTub) and xC < rBor - (rTub + eTub),
  "Parameters do not correspond to physically possible location of borehole tube.
    xC   = " + String(xC) + "
    rTub = " + String(rTub) + "
    eTub = " + String(eTub) + ".");
  R := Modelica.Math.log((2*P1*(P2^2 + 1)^P3)/(P2*(P2^2 - 1)^P3))/
           (Modelica.Constants.pi*kFil*hSeg);
end boreholeInternalResistance;

Buildings.Fluid.HeatExchangers.Boreholes.BaseClasses.boreholeThermalResistance

Information

This model computes the borehole total thermal resistance for a single U-tube. The computation is as defined in Hellstom (1991). The total thermal resistance is the contribution of three different resistances as

R_tot= R_Con + R_tub + R_fil.

where R_Con is the convective resistance of the fluid, R_tub is the conductive resistance of the tube wall and R_fil is the resistance of the filling material.

The convective resistance is obtained as

R_Con = 1 ⁄ (4 π h_seg r_tub h_in),

where h_seg is the height of the segment and h_in is the convective heat transfer coefficient. The convective heat transfer coefficient is determined using the Dittus-Boelter correlation,

h_in= 0.023 k_med Pr^0.35 Re^0.8 ⁄ 2 r_tub,

where k_med is the thermal conductivity of the fluid, Pr is the Prandtl number, Re is the Reynolds number and r_tub is the inside radius of the pipe.

The conductive resistance is determined using

R_tub= ln[ ( r_tub+e_tub ) ⁄ r_tub ] ⁄ 4 π k_tub h_seg,

where k_tub the heat conductivity of the pipe and e_tub is the thickness of the wall.

The resistance of the filling material is determined using the correlation from Paul (1996) :

R_fil = k_fil h_seg β₀ (r_Bor ⁄ (r_tub + e_tub)) ^β₁ ,

where k_fil is the thermal conductivity of the filling material and β₀, β₁ are the resistance shape factors (Paul 1996) based on U-tube shank spacing. Paul's shape factors are based on experimental and finite element analysis of typical borehole. The default values used for these coefficients are β₀= 20.100 and β₁=-0.94467. Values listed by Paul are given in the table below.

References

Hellstrom, G (1991).
Thermal Performance of Borehole Heat Exchangers.
Department of Mathematical Physics, Lund Institute of Technology.

Paul, N.D (1996). The effect of grout thermal conductivity on vertical geothermal heat exchanger design and performance. Master of Science Thesis, South Dakota State University.

Inputs

Type	Name	Default	Description
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]
ThermalConductivity	kTub		Thermal conductivity of the tubes [W/(m.K)]
ThermalConductivity	kFil		Thermal conductivity of the filling material [W/(m.K)]
ThermalConductivity	kMed		Thermal conductivity of the fluid [W/(m.K)]
DynamicViscosity	mueMed		Dynamic viscosity of the fluid [Pa.s]
SpecificHeatCapacity	cpFluid		Specific heat capacity of the fluid [J/(kg.K)]
MassFlowRate	m_flow		Mass flow rate of the fluid [kg/s]
MassFlowRate	m_flow_nominal		Nominal mass flow rate [kg/s]
Real	B0		Shape coefficient for grout resistance
Real	B1		Shape coefficient for grout resistance

Outputs

Type	Name	Description
ThermalResistance	Rb	Borehole thermal resistance [K/W]

Modelica definition

function boreholeThermalResistance 
  "Calculate the total thermal resistance of the borehole from the fluid to the soil"
  input Modelica.SIunits.Height hSeg "Height of the element";
  input Modelica.SIunits.Radius rBor "Radius of the borehole";
  input Modelica.SIunits.Radius rTub "Radius of the tube";
  input Modelica.SIunits.Length eTub "Thickness of the tubes";
  input Modelica.SIunits.ThermalConductivity kTub 
    "Thermal conductivity of the tubes";
  input Modelica.SIunits.ThermalConductivity kFil 
    "Thermal conductivity of the filling material";
  input Modelica.SIunits.ThermalConductivity kMed 
    "Thermal conductivity of the fluid";
  input Modelica.SIunits.DynamicViscosity mueMed 
    "Dynamic viscosity of the fluid";
  input Modelica.SIunits.SpecificHeatCapacity cpFluid 
    "Specific heat capacity of the fluid";
  input Modelica.SIunits.MassFlowRate m_flow "Mass flow rate of the fluid";
  input Modelica.SIunits.MassFlowRate m_flow_nominal "Nominal mass flow rate";
  input Real B0 "Shape coefficient for grout resistance";
  input Real B1 "Shape coefficient for grout resistance";
  output Modelica.SIunits.ThermalResistance Rb "Borehole thermal resistance";
protected 
  Modelica.SIunits.CoefficientOfHeatTransfer h 
    "Convective heat transfer coefficient of the fluid";
  Modelica.SIunits.ThermalResistance RCon 
    "Convective thermal resistance of the fluid";
  Modelica.SIunits.ThermalResistance RTub 
    "Thermal resistance of the pipe's wall";
  Modelica.SIunits.ThermalResistance RFil 
    "Thermal resistance of the filling material";
  Real k(unit="s/kg") 
    "Coefficient used in the computation of the convective heat transfer coefficient";
algorithm 
  k :=2/(mueMed*Modelica.Constants.pi*rTub);
  h := 0.023*kMed*(cpFluid*mueMed/kMed)^(0.35)/(2*rTub) *
    Buildings.Utilities.Math.Functions.regNonZeroPower(
    x=m_flow*k, n=0.8,
    delta=0.01*m_flow_nominal*k);
  RCon := 1/(4*Modelica.Constants.pi*rTub*hSeg*h);
  RTub := Modelica.Math.log((rTub + eTub)/rTub)/(4*Modelica.Constants.pi*hSeg*kTub);
  RFil := 1/(hSeg*kFil*B0*(rBor/(rTub + eTub))^B1);
  Rb := RCon + RTub + RFil;
end boreholeThermalResistance;

Buildings.Fluid.HeatExchangers.Boreholes.BaseClasses.exchangeValues

Information

External function that stores the input value x as the element a[iX] in the array a = [a[1], a[2], ...], and that returns the element a[iY]. The size of the array a is automatically enlarged as needed.

Inputs

Type	Name	Default	Description
ExtendableArray	table		External object
Integer	iX		One-based index where u needs to be stored in the array of the external object
Real	x		Value to store in the external object
Integer	iY		One-based index of the element that needs to be returned from C to Modelica

Outputs

Type	Name	Description
Real	y	Value of the i-th element

Modelica definition

function exchangeValues 
  "Store the value u at the element n in the external object, and return the value of the element i"
  input ExtendableArray table "External object";
  input Integer iX 
    "One-based index where u needs to be stored in the array of the external object";
  input Real x "Value to store in the external object";
  input Integer iY 
    "One-based index of the element that needs to be returned from C to Modelica";
  output Real y "Value of the i-th element";
external"C" y = exchangeValues(table, iX, x, iY);
end exchangeValues;

Buildings.Fluid.HeatExchangers.Boreholes.BaseClasses.factorial

Information

This function computes y = j!.

Inputs

Type	Name	Default	Description
Integer	j

Outputs

Type	Name	Description
Integer	f

Modelica definition

function factorial "Calculates the factorial value of the input"
  input Integer j;
  output Integer f;
algorithm 
  f := 1;
  for i in 1:j loop
    f := f*i;
  end for;
end factorial;

Buildings.Fluid.HeatExchangers.Boreholes.BaseClasses.powerSeries

Information

This function computes the power series that is used to compute the far-field temperature.

Inputs

Type	Name	Default	Description
Real	u		u
Integer	N		Number of coefficients

Outputs

Type	Name	Description
Real	W	Power series

Modelica definition

function powerSeries 
  "Power series used to compute far-field temperature"
  input Real u "u";
  input Integer N "Number of coefficients";
  output Real W "Power series";
algorithm 
  W := -0.5772 - Modelica.Math.log(u) + sum((-1)^(j + 1)*u^j/(j*factorial(j))
    for j in 1:N);
end powerSeries;

Buildings.Fluid.HeatExchangers.Boreholes.BaseClasses.temperatureDrop

Information

This function calculates the temperature drop of the soil at the outer boundary of the cylinder. The analytical formula of Hart and Couvillion (1986) for constant heat extraction is adapted to a non-constant heat flux. To adapt the formula for a variable rate of heat extraction, different constant heat extraction rates, starting at different time instances, are super-imposed. To obtain the temperature drop at the time t=n*Δt, the effects of constant rate of heat extractions are super-imposed as

ΔT ( r , t=n Δt )= 1 ⁄ ( 4 π k ) ∑ W(u(r, t= i Δt)) (q_n-i+1-q_n-i),

where r is the radius for which the temperature is computed, k is the thermal conductivity of the material, W is a solution of the heat conduction in polar coordinates and q_i=Q_i/h is the specific rate of heat extraction per unit lenght at time t=i Δt. The value of W is obtained using

W(u)=[-0.5772 - ln(u) + u - u²/(2   2!) +u³/(3   3!) - u⁴/(4   4!) + ....].

where u(r,t)= c ρ r² ⁄ (4 t k) , ρ is the mass density and c is the specific heat capacity per unit mass.

Implementation

The rate of heat flow Q_i is obtained from the function Buildings.Fluid.HeatExchangers.Boreholes.BaseClasses.exchangeValues.

References

Hart and Couvillion, (1986). Earth Coupled Heat Transfer. Publication of the National Water Well Association.

Inputs

Type	Name	Default	Description
ExtendableArray	table		External object that contains the history terms of the heat flux
Integer	iSam		Counter for how many time the model was sampled. Defined as iSam=1 when called at t=0
HeatFlowRate	Q_flow		Heat flow rate to be stored in the external object [W]
Time	samplePeriod		Period between two samples [s]
Radius	rExt		External radius of the cylinder [m]
Height	hSeg		Height of the cylinder [m]
ThermalConductivity	k		Thermal conductivity of the soil [W/(m.K)]
Density	d		Density of the soil [kg/m3]
SpecificHeatCapacity	c		Specific heat capacity of the soil [J/(kg.K)]

Outputs

Type	Name	Description
TemperatureDifference	dT	Temperature drop of the soil [K]

Modelica definition

function temperatureDrop 
  "Calculate the temperature drop of the soil at the external boundary of the cylinder"
input ExtendableArray table 
    "External object that contains the history terms of the heat flux";
input Integer iSam(min=1) 
    "Counter for how many time the model was sampled. Defined as iSam=1 when called at t=0";

input Modelica.SIunits.HeatFlowRate Q_flow 
    "Heat flow rate to be stored in the external object";
input Modelica.SIunits.Time samplePeriod "Period between two samples";
input Modelica.SIunits.Radius rExt "External radius of the cylinder";
input Modelica.SIunits.Height hSeg "Height of the cylinder";
input Modelica.SIunits.ThermalConductivity k "Thermal conductivity of the soil";
input Modelica.SIunits.Density d "Density of the soil";
input Modelica.SIunits.SpecificHeatCapacity c 
    "Specific heat capacity of the soil";
output Modelica.SIunits.TemperatureDifference dT "Temperature drop of the soil";
protected 
 Modelica.SIunits.Time minSamplePeriod= rExt^2/(4*(k/c/d)*3.8) 
    "Minimal lenght of the sampling period";
 Modelica.SIunits.HeatFlowRate QL_flow 
    "Intermediate variable for heat flow rate at the lower bound of the time interval";
 Modelica.SIunits.HeatFlowRate QU_flow 
    "Intermediate variable for heat flow rate at the upper bound of the time interval";

algorithm 
  assert(rExt*rExt/(4*(k/c/d)*samplePeriod)<=3.8,
  "The samplePeriod has to be bigger than " + String(minSamplePeriod) + " for convergence purpose.
  samplePeriod = "+ String(samplePeriod));
  if iSam == 1 then
    // First call, at t=0
    dT := 0;
    QL_flow := Buildings.Fluid.HeatExchangers.Boreholes.BaseClasses.exchangeValues(
                       table=table, iX=iSam, x=Q_flow, iY=iSam);
  else
    dT := 0;
    // The first evaluation is at iSam=2, in which case we have one term of the sum,
    // and t=samplePeriod=(iSam-1)*samplePeriod
   for i in 1:(iSam-1) loop
      QL_flow := Buildings.Fluid.HeatExchangers.Boreholes.BaseClasses.exchangeValues(
        table=table, iX=iSam, x=Q_flow, iY=iSam+1-i);
      QU_flow := Buildings.Fluid.HeatExchangers.Boreholes.BaseClasses.exchangeValues(
        table=table, iX=iSam, x=Q_flow, iY=iSam-i);
      // The division by hSeg is because QU_flow and QL_flow are in [W], but the equation
      // requires [W/m], i.e., heat flow rate per unit length of the line source.
      dT := dT + 1/(4*Modelica.Constants.pi*k)*
        Buildings.Fluid.HeatExchangers.Boreholes.BaseClasses.powerSeries(
           u=c*d/(4*k*i*samplePeriod)*rExt^2, N=10)*
        (QL_flow-QU_flow)/hSeg;
 end for;
  end if;
end temperatureDrop;