Buildings.Fluid.Geothermal.Borefields.BaseClasses.HeatTransfer.ThermalResponseFactors

Models for heat transfer outside boreholes

Information

This package contains functions to evaluate temperature response factors used by Buildings.Fluid.Geothermal.Borefields.BaseClasses.HeatTransfer.GroundTemperatureResponse to evaluate borehole wall temperatures.

Extends from Modelica.Icons.BasesPackage (Icon for packages containing base classes).

Package Content

Name	Description
clusterBoreholes	Identify clusters of boreholes with similar heat interactions
cylindricalHeatSource	Cylindrical heat source solution from Carslaw and Jaeger
cylindricalHeatSource_Integrand	Integrand function for cylindrical heat source evaluation
finiteLineSource	Finite line source solution of Claesson and Javed
finiteLineSource_Equivalent	Finite line source solution of Prieto and Cimmino
finiteLineSource_Erfint	Integral of the error function
finiteLineSource_Integrand	Integrand function for finite line source evaluation
finiteLineSource_Integrand_Equivalent	Integrand function for finite line source evaluation
finiteLineSource_SteadyState	Steady-state finite line source solution
gFunction	Evaluate the g-function of a bore field
infiniteLineSource	Infinite line source model for borehole heat exchangers
shaGFunction	Returns a SHA1 encryption of the formatted arguments for the g-function generation
timeGeometric	Geometric expansion of time steps
Validation	Example models for ThermalResponseFactors

Buildings.Fluid.Geothermal.Borefields.BaseClasses.HeatTransfer.ThermalResponseFactors.clusterBoreholes

Identify clusters of boreholes with similar heat interactions

Information

This function identifies groups of similarly behaving boreholes using a k-means clustering algorithm. Boreholes are clustered based on their steady-state dimensionless borehole wall temperatures obtained from the spatial superposition of the steady-state finite line source solution (see Buildings.Fluid.Geothermal.Borefields.BaseClasses.HeatTransfer.ThermalResponseFactors.finiteLineSource_SteadyState).

Implementation

The implemented method differs from the method presented by Prieto and Cimmino (2021). They used a hierarchical agglomerative clustering method with complete linkage to identify the borehole clusters. The optimal number of clusters was identified by cutting the dendrogram generated during the clustering process.

Here, a k-means algorithm is used instead, using the euclidian distance between steady-state borehole wall temperatures. The number of clusters is a parameter in this approach. However, as observed by Prieto and Cimmino (2021), nClu=5 clusters should provide acceptable accuracy in most practical cases. This number can be increased without significant change in the computational cost.

References

Prieto, C. and Cimmino, M. 2021. Thermal interactions in large irregular fields of geothermal boreholes: the method of equivalent boreholes. Journal of Building Performance Simulation 14(4): 446-460. doi:10.1080/19401493.2021.1968953.

Extends from Modelica.Icons.Function (Icon for functions).

Inputs

Type	Name	Default	Description
Integer	nBor		Number of boreholes
Position	cooBor[nBor, 2]		Coordinates of boreholes [m]
Height	hBor		Borehole length [m]
Height	dBor		Borehole buried depth [m]
Radius	rBor		Borehole radius [m]
Integer	nClu		Number of clusters to be generated
TemperatureDifference	TTol	0.001	Absolute tolerance on the borehole wall temperature for the identification of clusters [K]

Outputs

Type	Name	Description
Integer	labels[nBor]	Cluster label associated with each data point
Integer	cluSiz[nClu]	Size of the clusters
Integer	N	Number of unique clusters

Modelica definition

impure function clusterBoreholes "Identify clusters of boreholes with similar heat interactions" extends Modelica.Icons.Function; input Integer nBor "Number of boreholes"; input Modelica.Units.SI.Position cooBor[nBor, 2] "Coordinates of boreholes"; input Modelica.Units.SI.Height hBor "Borehole length"; input Modelica.Units.SI.Height dBor "Borehole buried depth"; input Modelica.Units.SI.Radius rBor "Borehole radius"; input Integer nClu "Number of clusters to be generated"; input Modelica.Units.SI.TemperatureDifference TTol = 0.001 "Absolute tolerance on the borehole wall temperature for the identification of clusters"; output Integer labels[nBor] "Cluster label associated with each data point"; output Integer cluSiz[nClu] "Size of the clusters"; output Integer N "Number of unique clusters"; protected Modelica.Units.SI.Temperature TBor[nBor,1] "Steady-state borehole wall temperatures"; Modelica.Units.SI.Temperature TBor_Unique[nBor] "Unique borehole wall temperatures under tolerance"; Modelica.Units.SI.Length dis "Distance between boreholes"; algorithm // ---- Evaluate borehole wall temperatures for i in 1:nBor loop TBor[i,1] := 0; for j in 1:nBor loop if i <> j then dis := sqrt((cooBor[i,1]-cooBor[j,1])^2 + (cooBor[i,2]-cooBor[j,2])^2); else dis := rBor; end if; TBor[i,1] := TBor[i,1] + Buildings.Fluid.Geothermal.Borefields.BaseClasses.HeatTransfer.ThermalResponseFactors.finiteLineSource_SteadyState(dis, hBor, dBor, hBor, dBor); end for; end for; // ---- Find all unique borehole wall temperatures under tolerance // The number of clusters is min(N, nClu) N := 1; TBor_Unique[1] := TBor[1,1]; if nClu > 1 and nBor > 1 then for i in 2:nBor loop for j in 1:N loop if abs(TBor[i,1] - TBor_Unique[j]) < TTol then break; elseif j == N then TBor_Unique[N+1] := TBor[i,1]; N := N + 1; end if; end for; end for; end if; N := min(N, nClu); // ---- Identify borehole clusters // This function is impure (,labels,cluSiz) := Buildings.Utilities.Clustering.KMeans( TBor, N, nBor, 1, n_cluster_size=nClu); end clusterBoreholes;

Buildings.Fluid.Geothermal.Borefields.BaseClasses.HeatTransfer.ThermalResponseFactors.cylindricalHeatSource

Cylindrical heat source solution from Carslaw and Jaeger

Information

This function evaluates the cylindrical heat source solution. This solution gives the relation between the constant heat transfer rate (per unit length) injected by a cylindrical heat source of infinite length and the temperature raise in the medium. The cylindrical heat source solution is defined by

where ΔT(t,r) is the temperature raise after a time t of constant heat injection and at a distance r from the cylindrical source, Q' is the heat injection rate per unit length, k_s is the soil thermal conductivity, Fo is the Fourier number, aSoi_s is the ground thermal diffusivity, r_b is the radius of the cylindrical source and G is the cylindrical heat source solution.

The cylindrical heat source solution is given by:

The integral is solved numerically, with the integrand defined in Buildings.Fluid.Geothermal.Borefields.BaseClasses.HeatTransfer.ThermalResponseFactors.cylindricalHeatSource_Integrand.

Extends from Modelica.Icons.Function (Icon for functions).

Inputs

Type	Name	Description
Time	t	Time [s]
ThermalDiffusivity	aSoi	Ground thermal diffusivity [m2/s]
Distance	dis	Radial distance between borehole axes [m]
Radius	rBor	Radius of emitting borehole [m]

Outputs

Type	Name	Description
Real	G	Thermal response factor of borehole 1 on borehole 2

Modelica definition

function cylindricalHeatSource "Cylindrical heat source solution from Carslaw and Jaeger" extends Modelica.Icons.Function; input Modelica.Units.SI.Time t "Time"; input Modelica.Units.SI.ThermalDiffusivity aSoi "Ground thermal diffusivity"; input Modelica.Units.SI.Distance dis "Radial distance between borehole axes"; input Modelica.Units.SI.Radius rBor "Radius of emitting borehole"; output Real G "Thermal response factor of borehole 1 on borehole 2"; protected Real Fo = aSoi*t/rBor^2 "Fourier number"; Real p = dis/rBor "Fourier number"; algorithm G := Modelica.Math.Nonlinear.quadratureLobatto( function Buildings.Fluid.Geothermal.Borefields.BaseClasses.HeatTransfer.ThermalResponseFactors.cylindricalHeatSource_Integrand( Fo=Fo, p=p), a = 1e-12, b = 100, tolerance = 1e-6); end cylindricalHeatSource;

Buildings.Fluid.Geothermal.Borefields.BaseClasses.HeatTransfer.ThermalResponseFactors.cylindricalHeatSource_Integrand

Integrand function for cylindrical heat source evaluation

Information

Integrand of the cylindrical heat source solution for use in Buildings.Fluid.Geothermal.Borefields.BaseClasses.HeatTransfer.ThermalResponseFactors.cylindricalHeatSource.

Extends from Modelica.Icons.Function (Icon for functions).

Inputs

Type	Name	Description
Real	u	Normalized integration variable
Real	Fo	Fourier number
Real	p	Ratio of distance over radius

Outputs

Type	Name	Description
Real	y	Value of integrand

Modelica definition

function cylindricalHeatSource_Integrand "Integrand function for cylindrical heat source evaluation" extends Modelica.Icons.Function; input Real u "Normalized integration variable"; input Real Fo "Fourier number"; input Real p "Ratio of distance over radius"; output Real y "Value of integrand"; algorithm y := 1.0/(u^2*Modelica.Constants.pi^2)*(exp(-u^2*Fo) - 1.0) /(Buildings.Utilities.Math.Functions.besselJ1(u)^2+Buildings.Utilities.Math.Functions.besselY1(u)^2) *(Buildings.Utilities.Math.Functions.besselJ0(p*u)*Buildings.Utilities.Math.Functions.besselY1(u) -Buildings.Utilities.Math.Functions.besselJ1(u)*Buildings.Utilities.Math.Functions.besselY0(p*u)); end cylindricalHeatSource_Integrand;

Buildings.Fluid.Geothermal.Borefields.BaseClasses.HeatTransfer.ThermalResponseFactors.finiteLineSource

Finite line source solution of Claesson and Javed

Information

This function evaluates the finite line source solution. This solution gives the relation between the constant heat transfer rate (per unit length) injected by a line source of finite length H₁ buried at a distance D₁ from a constant temperature surface (T=0) and the average temperature raise over a line of finite length H₂ buried at a distance D₂ from the constant temperature surface. The finite line source solution is defined by:

where ΔT_1-2(t,r,H₁,D₁,H₂,D₂) is the temperature raise after a time t of constant heat injection and at a distance r from the line heat source, Q' is the heat injection rate per unit length, k_s is the soil thermal conductivity and h_FLS is the finite line source solution.

The finite line source solution is given by:

where α_s is the ground thermal diffusivity and erfint is the integral of the error function, defined in Buildings.Fluid.Geothermal.Borefields.BaseClasses.HeatTransfer.ThermalResponseFactors.finiteLineSource_erfint. The integral is solved numerically, with the integrand defined in Buildings.Fluid.Geothermal.Borefields.BaseClasses.HeatTransfer.ThermalResponseFactors.finiteLineSource_Integrand.

Extends from Modelica.Icons.Function (Icon for functions).

Inputs

Type	Name	Default	Description
Time	t		Time [s]
ThermalDiffusivity	aSoi		Ground thermal diffusivity [m2/s]
Distance	dis		Radial distance between borehole axes [m]
Height	len1		Length of emitting borehole [m]
Height	burDep1		Buried depth of emitting borehole [m]
Height	len2		Length of receiving borehole [m]
Height	burDep2		Buried depth of receiving borehole [m]
Boolean	includeRealSource	true	True if contribution of real source is included
Boolean	includeMirrorSource	true	True if contribution of mirror source is included

Outputs

Type	Name	Description
Real	h_21	Thermal response factor of borehole 1 on borehole 2

Modelica definition

function finiteLineSource "Finite line source solution of Claesson and Javed" extends Modelica.Icons.Function; input Modelica.Units.SI.Time t "Time"; input Modelica.Units.SI.ThermalDiffusivity aSoi "Ground thermal diffusivity"; input Modelica.Units.SI.Distance dis "Radial distance between borehole axes"; input Modelica.Units.SI.Height len1 "Length of emitting borehole"; input Modelica.Units.SI.Height burDep1 "Buried depth of emitting borehole"; input Modelica.Units.SI.Height len2 "Length of receiving borehole"; input Modelica.Units.SI.Height burDep2 "Buried depth of receiving borehole"; input Boolean includeRealSource = true "True if contribution of real source is included"; input Boolean includeMirrorSource = true "True if contribution of mirror source is included"; output Real h_21 "Thermal response factor of borehole 1 on borehole 2"; protected Real lowBou(unit="m-1") "Lower bound of integration"; // Upper bound is infinite Real uppBou(unit="m-1") = max(100.0, 10.0/dis) "Upper bound of integration"; Modelica.Units.SI.Distance disMin "Minimum distance between sources and receiving line"; Modelica.Units.SI.Time timTre "Time treshold for evaluation of the solution"; algorithm h_21 := 0; if t > 0 and (includeRealSource or includeMirrorSource) then // Find the minimum distance between the line source and the line where the // finite line source solution is evaluated. if includeRealSource then if (burDep2 + len2) < burDep1 then disMin := sqrt(dis^2 + (burDep1 - burDep2 - len2)^2); elseif burDep2 > (burDep1 + len1) then disMin := sqrt(dis^2 + (burDep1 - burDep2 + len1)^2); else disMin := dis; end if; else disMin := sqrt(dis^2 + (burDep1 + burDep2)^2); end if; // The traveled distance of the temperature front is assumed to be: // d = 5*sqrt(aSoi*t). // The solution is only evaluated at times when the traveled distance is // greater than the minimum distance. timTre := disMin^2/(25*aSoi); if t >= timTre then lowBou := 1.0/sqrt(4*aSoi*t); h_21 := Modelica.Math.Nonlinear.quadratureLobatto( function Buildings.Fluid.Geothermal.Borefields.BaseClasses.HeatTransfer.ThermalResponseFactors.finiteLineSource_Integrand( dis=dis, len1=len1, burDep1=burDep1, len2=len2, burDep2=burDep2, includeRealSource=includeRealSource, includeMirrorSource=includeMirrorSource), lowBou, uppBou, 1.0e-6); else // Linearize the solution at times below the time treshold. lowBou := 1.0/sqrt(4*aSoi*timTre); h_21 := t/timTre*Modelica.Math.Nonlinear.quadratureLobatto( function Buildings.Fluid.Geothermal.Borefields.BaseClasses.HeatTransfer.ThermalResponseFactors.finiteLineSource_Integrand( dis=dis, len1=len1, burDep1=burDep1, len2=len2, burDep2=burDep2, includeRealSource=includeRealSource, includeMirrorSource=includeMirrorSource), lowBou, uppBou, 1.0e-6); end if; end if; end finiteLineSource;

Buildings.Fluid.Geothermal.Borefields.BaseClasses.HeatTransfer.ThermalResponseFactors.finiteLineSource_Equivalent

Finite line source solution of Prieto and Cimmino

Information

This function evaluates the finite line source solution. This solution gives the relation between the constant heat transfer rate (per unit length) injected by a group of line sources of finite length H₁ buried at a distance D₁ from a constant temperature surface (T=0) and the average temperature raise over a group of N₂ lines of finite length H₂ buried at a distance D₂ from the constant temperature surface. The finite line source solution is defined by:

where ΔT_1-2(t,r_1->2,H₁,D₁,H₂,D₂,N₂) is the temperature raise over lines in group 2 after a time t of constant heat injection, r_1->2 is the list of distance between all pairs of lines in groups 1 and 2, N_1->2 is the number of distances, Q' is the heat injection rate per unit length, k_s is the soil thermal conductivity and h_FLS is the finite line source solution.

The finite line source solution is given by:

Extends from Modelica.Icons.Function (Icon for functions).

Inputs

Type	Name	Default	Description
Time	t		Time [s]
ThermalDiffusivity	aSoi		Ground thermal diffusivity [m2/s]
Distance	dis[n_dis]		Radial distance between borehole axes [m]
Integer	wDis[n_dis]		Number of occurences of each distance
Height	len1		Length of emitting boreholes [m]
Height	burDep1		Buried depth of emitting boreholes [m]
Height	len2		Length of receiving boreholes [m]
Height	burDep2		Buried depth of receiving boreholes [m]
Integer	nBor2		Number of receiving boreholes
Integer	n_dis		Number of unique distances
Boolean	includeRealSource	true	True if contribution of real source is included
Boolean	includeMirrorSource	true	True if contribution of mirror source is included

Outputs

Type	Name	Description
Real	h_21	Thermal response factor of borehole 1 on borehole 2

Modelica definition

function finiteLineSource_Equivalent "Finite line source solution of Prieto and Cimmino" extends Modelica.Icons.Function; input Modelica.Units.SI.Time t "Time"; input Modelica.Units.SI.ThermalDiffusivity aSoi "Ground thermal diffusivity"; input Modelica.Units.SI.Distance dis[n_dis] "Radial distance between borehole axes"; input Integer wDis[n_dis] "Number of occurences of each distance"; input Modelica.Units.SI.Height len1 "Length of emitting boreholes"; input Modelica.Units.SI.Height burDep1 "Buried depth of emitting boreholes"; input Modelica.Units.SI.Height len2 "Length of receiving boreholes"; input Modelica.Units.SI.Height burDep2 "Buried depth of receiving boreholes"; input Integer nBor2 "Number of receiving boreholes"; input Integer n_dis "Number of unique distances"; input Boolean includeRealSource = true "True if contribution of real source is included"; input Boolean includeMirrorSource = true "True if contribution of mirror source is included"; output Real h_21 "Thermal response factor of borehole 1 on borehole 2"; protected Real lowBou(unit="m-1") "Lower bound of integration"; // Upper bound is infinite Real uppBou(unit="m-1") = max(100.0, 10.0/min(dis)) "Upper bound of integration"; Modelica.Units.SI.Distance disMin "Minimum distance between sources and receiving line"; Modelica.Units.SI.Time timTre "Time treshold for evaluation of the solution"; algorithm h_21 := 0; if t > 0 and (includeRealSource or includeMirrorSource) then // Find the minimum distance between the line source and the line where the // finite line source solution is evaluated. if includeRealSource then if (burDep2 + len2) < burDep1 then disMin := sqrt(min(dis)^2 + (burDep1 - burDep2 - len2)^2); elseif burDep2 > (burDep1 + len1) then disMin := sqrt(min(dis)^2 + (burDep1 - burDep2 + len1)^2); else disMin := min(dis); end if; else disMin := sqrt(min(dis)^2 + (burDep1 + burDep2)^2); end if; // The traveled distance of the temperature front is assumed to be: // d = 5*sqrt(aSoi*t). // The solution is only evaluated at times when the traveled distance is // greater than the minimum distance. timTre := disMin^2/(25*aSoi); if t >= timTre then lowBou := 1.0/sqrt(4*aSoi*t); h_21 :=Modelica.Math.Nonlinear.quadratureLobatto( function Buildings.Fluid.Geothermal.Borefields.BaseClasses.HeatTransfer.ThermalResponseFactors.finiteLineSource_Integrand_Equivalent( dis=dis, wDis=wDis, len1=len1, burDep1=burDep1, len2=len2, burDep2=burDep2, nBor2=nBor2, n_dis=n_dis, includeRealSource=includeRealSource, includeMirrorSource=includeMirrorSource), lowBou, uppBou, 1.0e-6); else // Linearize the solution at times below the time treshold. lowBou := 1.0/sqrt(4*aSoi*timTre); h_21 :=t/timTre*Modelica.Math.Nonlinear.quadratureLobatto( function Buildings.Fluid.Geothermal.Borefields.BaseClasses.HeatTransfer.ThermalResponseFactors.finiteLineSource_Integrand_Equivalent( dis=dis, wDis=wDis, len1=len1, burDep1=burDep1, len2=len2, burDep2=burDep2, nBor2=nBor2, n_dis=n_dis, includeRealSource=includeRealSource, includeMirrorSource=includeMirrorSource), lowBou, uppBou, 1.0e-6); end if; end if; end finiteLineSource_Equivalent;

Buildings.Fluid.Geothermal.Borefields.BaseClasses.HeatTransfer.ThermalResponseFactors.finiteLineSource_Erfint

Integral of the error function

Information

This function evaluates the integral of the error function, given by:

Extends from Modelica.Math.Nonlinear.Interfaces.partialScalarFunction (Interface for a function with one input and one output Real signal).

Inputs

Type	Name	Default	Description
Real	u		Independent variable

Outputs

Type	Name	Description
Real	y	Dependent variable y=f(u)

Modelica definition

function finiteLineSource_Erfint "Integral of the error function" extends Modelica.Math.Nonlinear.Interfaces.partialScalarFunction; algorithm y := u*Modelica.Math.Special.erf(u) - 1/sqrt(Modelica.Constants.pi)*(1 - exp(-u^2)); end finiteLineSource_Erfint;

Buildings.Fluid.Geothermal.Borefields.BaseClasses.HeatTransfer.ThermalResponseFactors.finiteLineSource_Integrand

Integrand function for finite line source evaluation

Information

Integrand of the finite line source solution for use in Buildings.Fluid.Geothermal.Borefields.BaseClasses.HeatTransfer.ThermalResponseFactors.finiteLineSource.

Extends from Modelica.Icons.Function (Icon for functions).

Inputs

Type	Name	Default	Description
Real	u		Integration variable [1/m]
Distance	dis		Radial distance between borehole axes [m]
Height	len1		Length of emitting borehole [m]
Height	burDep1		Buried depth of emitting borehole [m]
Height	len2		Length of receiving borehole [m]
Height	burDep2		Buried depth of receiving borehole [m]
Boolean	includeRealSource	true	true if contribution of real source is included
Boolean	includeMirrorSource	true	true if contribution of mirror source is included

Outputs

Type	Name	Description
Real	y	Value of integrand [m]

Modelica definition

function finiteLineSource_Integrand "Integrand function for finite line source evaluation" extends Modelica.Icons.Function; input Real u(unit="1/m") "Integration variable"; input Modelica.Units.SI.Distance dis "Radial distance between borehole axes"; input Modelica.Units.SI.Height len1 "Length of emitting borehole"; input Modelica.Units.SI.Height burDep1 "Buried depth of emitting borehole"; input Modelica.Units.SI.Height len2 "Length of receiving borehole"; input Modelica.Units.SI.Height burDep2 "Buried depth of receiving borehole"; input Boolean includeRealSource = true "true if contribution of real source is included"; input Boolean includeMirrorSource = true "true if contribution of mirror source is included"; output Real y(unit="m") "Value of integrand"; protected Real f "Intermediate variable"; algorithm if includeRealSource then f := sum({ +Buildings.Fluid.Geothermal.Borefields.BaseClasses.HeatTransfer.ThermalResponseFactors.finiteLineSource_Erfint( (burDep2 - burDep1 + len2)*u), -Buildings.Fluid.Geothermal.Borefields.BaseClasses.HeatTransfer.ThermalResponseFactors.finiteLineSource_Erfint( (burDep2 - burDep1)*u), +Buildings.Fluid.Geothermal.Borefields.BaseClasses.HeatTransfer.ThermalResponseFactors.finiteLineSource_Erfint( (burDep2 - burDep1 - len1)*u), -Buildings.Fluid.Geothermal.Borefields.BaseClasses.HeatTransfer.ThermalResponseFactors.finiteLineSource_Erfint( (burDep2 - burDep1 + len2 - len1)*u)}); else f := 0; end if; if includeMirrorSource then f := f + sum({ +Buildings.Fluid.Geothermal.Borefields.BaseClasses.HeatTransfer.ThermalResponseFactors.finiteLineSource_Erfint( (burDep2 + burDep1 + len2)*u), -Buildings.Fluid.Geothermal.Borefields.BaseClasses.HeatTransfer.ThermalResponseFactors.finiteLineSource_Erfint( (burDep2 + burDep1)*u), +Buildings.Fluid.Geothermal.Borefields.BaseClasses.HeatTransfer.ThermalResponseFactors.finiteLineSource_Erfint( (burDep2 + burDep1 + len1)*u), -Buildings.Fluid.Geothermal.Borefields.BaseClasses.HeatTransfer.ThermalResponseFactors.finiteLineSource_Erfint( (burDep2 + burDep1 + len2 + len1)*u)}); end if; y := 0.5/(len2*u^2)*f*exp(-dis^2*u^2); end finiteLineSource_Integrand;

Buildings.Fluid.Geothermal.Borefields.BaseClasses.HeatTransfer.ThermalResponseFactors.finiteLineSource_Integrand_Equivalent

Integrand function for finite line source evaluation

Information

Integrand of the finite line source solution for use in Buildings.Fluid.Geothermal.Borefields.BaseClasses.HeatTransfer.ThermalResponseFactors.finiteLineSource_Equivalent.

Extends from Modelica.Icons.Function (Icon for functions).

Inputs

Type	Name	Default	Description
Real	u		Integration variable [1/m]
Distance	dis[n_dis]		Radial distance between borehole axes [m]
Integer	wDis[n_dis]
Height	len1		Length of emitting boreholes [m]
Height	burDep1		Buried depth of emitting boreholes [m]
Height	len2		Length of receiving boreholes [m]
Height	burDep2		Buried depth of receiving boreholes [m]
Integer	nBor2		Number of receiving boreholes over which the response is averaged
Integer	n_dis		Number of unique distances between emitting and receiving boreholes
Boolean	includeRealSource	true	true if contribution of real source is included
Boolean	includeMirrorSource	true	true if contribution of mirror source is included

Outputs

Type	Name	Description
Real	y	Value of integrand [m]

Modelica definition

function finiteLineSource_Integrand_Equivalent "Integrand function for finite line source evaluation" extends Modelica.Icons.Function; input Real u(unit="1/m") "Integration variable"; input Modelica.Units.SI.Distance dis[n_dis] "Radial distance between borehole axes"; input Integer wDis[n_dis]; input Modelica.Units.SI.Height len1 "Length of emitting boreholes"; input Modelica.Units.SI.Height burDep1 "Buried depth of emitting boreholes"; input Modelica.Units.SI.Height len2 "Length of receiving boreholes"; input Modelica.Units.SI.Height burDep2 "Buried depth of receiving boreholes"; input Integer nBor2 "Number of receiving boreholes over which the response is averaged"; input Integer n_dis "Number of unique distances between emitting and receiving boreholes"; input Boolean includeRealSource = true "true if contribution of real source is included"; input Boolean includeMirrorSource = true "true if contribution of mirror source is included"; output Real y(unit="m") "Value of integrand"; protected Real f "Intermediate variable"; algorithm if includeRealSource then f := sum({ +Buildings.Fluid.Geothermal.Borefields.BaseClasses.HeatTransfer.ThermalResponseFactors.finiteLineSource_Erfint( (burDep2 - burDep1 + len2)*u), -Buildings.Fluid.Geothermal.Borefields.BaseClasses.HeatTransfer.ThermalResponseFactors.finiteLineSource_Erfint( (burDep2 - burDep1)*u), +Buildings.Fluid.Geothermal.Borefields.BaseClasses.HeatTransfer.ThermalResponseFactors.finiteLineSource_Erfint( (burDep2 - burDep1 - len1)*u), -Buildings.Fluid.Geothermal.Borefields.BaseClasses.HeatTransfer.ThermalResponseFactors.finiteLineSource_Erfint( (burDep2 - burDep1 + len2 - len1)*u)}); else f := 0; end if; if includeMirrorSource then f := f + sum({ +Buildings.Fluid.Geothermal.Borefields.BaseClasses.HeatTransfer.ThermalResponseFactors.finiteLineSource_Erfint( (burDep2 + burDep1 + len2)*u), -Buildings.Fluid.Geothermal.Borefields.BaseClasses.HeatTransfer.ThermalResponseFactors.finiteLineSource_Erfint( (burDep2 + burDep1)*u), +Buildings.Fluid.Geothermal.Borefields.BaseClasses.HeatTransfer.ThermalResponseFactors.finiteLineSource_Erfint( (burDep2 + burDep1 + len1)*u), -Buildings.Fluid.Geothermal.Borefields.BaseClasses.HeatTransfer.ThermalResponseFactors.finiteLineSource_Erfint( (burDep2 + burDep1 + len2 + len1)*u)}); end if; y := 0.5/(nBor2*len2*u^2)*f* (wDis*exp(-dis.*dis*u^2)); end finiteLineSource_Integrand_Equivalent;

Buildings.Fluid.Geothermal.Borefields.BaseClasses.HeatTransfer.ThermalResponseFactors.finiteLineSource_SteadyState

Steady-state finite line source solution

Information

This function evaluates the steady-state value of finite line source solution. This solution gives the relation between the constant heat transfer rate (per unit length) injected by a line source of finite length H₁ buried at a distance D₁ from a constant temperature surface (T=0) and the average temperature raise over a line of finite length H₂ buried at a distance D₂ from the constant temperature surface. The finite line source solution is defined by:

The steady-state finite line source solution is given by:

Extends from Modelica.Icons.Function (Icon for functions).

Inputs

Type	Name	Default	Description
Distance	dis		Radial distance between borehole axes [m]
Height	len1		Length of emitting borehole [m]
Height	burDep1		Buried depth of emitting borehole [m]
Height	len2		Length of receiving borehole [m]
Height	burDep2		Buried depth of receiving borehole [m]
Boolean	includeRealSource	true	True if contribution of real source is included
Boolean	includeMirrorSource	true	True if contribution of mirror source is included

Outputs

Type	Name	Description
Real	h_21	Thermal response factor of borehole 1 on borehole 2

Modelica definition

function finiteLineSource_SteadyState "Steady-state finite line source solution" extends Modelica.Icons.Function; input Modelica.Units.SI.Distance dis "Radial distance between borehole axes"; input Modelica.Units.SI.Height len1 "Length of emitting borehole"; input Modelica.Units.SI.Height burDep1 "Buried depth of emitting borehole"; input Modelica.Units.SI.Height len2 "Length of receiving borehole"; input Modelica.Units.SI.Height burDep2 "Buried depth of receiving borehole"; input Boolean includeRealSource = true "True if contribution of real source is included"; input Boolean includeMirrorSource = true "True if contribution of mirror source is included"; output Real h_21 "Thermal response factor of borehole 1 on borehole 2"; protected Real f "Intermediate variable for FLS solution"; Real p[4] = {1., -1., 1., -1.} "Intermediate variable for sum coefficients"; Real q[4] "Intermediate variable for axial distances"; algorithm if includeRealSource then q := {burDep2 - burDep1 + len2, burDep2 - burDep1, burDep2 - burDep1 - len1, burDep2 - burDep1 + len2 - len1}; f := p * (q .* log((q + sqrt(q.*q + ones(4)*dis^2)) / dis) - sqrt(q.*q + ones(4)*dis^2)); else f := 0; end if; if includeMirrorSource then q := {burDep2 + burDep1 + len2, burDep2 + burDep1, burDep2 + burDep1 + len1, burDep2 + burDep1 + len2 + len1}; f := f + p * (q .* log((q + sqrt(q.*q + ones(4)*dis^2)) / dis) - sqrt(q.*q + ones(4)*dis^2)); end if; h_21 := 0.5 * f / len2; end finiteLineSource_SteadyState;

Buildings.Fluid.Geothermal.Borefields.BaseClasses.HeatTransfer.ThermalResponseFactors.gFunction

Evaluate the g-function of a bore field

Information

This function implements the g-function evaluation method introduced by Cimmino and Bernier (see: Cimmino and Bernier (2014), Cimmino (2018), and Prieto and Cimmino (2021)) based on the g-function function concept first introduced by Eskilson (1987). The g-function gives the relation between the variation of the borehole wall temperature at a time t and the heat extraction and injection rates at all times preceding time t as

where T_b is the borehole wall temperature, T_g is the undisturbed ground temperature, Q is the heat injection rate into the ground through the borehole wall per unit borehole length, k_s is the soil thermal conductivity and g is the g-function.

The g-function is constructed from the combination of the combination of the finite line source (FLS) solution (see Buildings.Fluid.Geothermal.Borefields.BaseClasses.HeatTransfer.ThermalResponseFactors.finiteLineSource), the cylindrical heat source (CHS) solution (see Buildings.Fluid.Geothermal.Borefields.BaseClasses.HeatTransfer.ThermalResponseFactors.cylindricalHeatSource), and the infinite line source (ILS) solution (see Buildings.Fluid.Geothermal.Borefields.BaseClasses.HeatTransfer.ThermalResponseFactors.infiniteLineSource). To obtain the g-function of a bore field, the bore field is first divided into nClu groups of similarly behaving boreholes. Each group is represented by a single equivalent borehole. Each equivalent borehole is then divided into a series of nSeg segments of equal length, each modeled as a line source of finite length. The finite line source solution is superimposed in space to obtain a system of equations that gives the relation between the heat injection rate at each of the segments and the borehole wall temperature at each of the segments. The system is solved to obtain the uniform borehole wall temperature required at any time to maintain a constant total heat injection rate (Q_tot = 2πk_sH_tot) into the bore field. The uniform borehole wall temperature is then equal to the finite line source based g-function.

Since this g-function is based on line sources of heat, rather than cylinders, the g-function is corrected to consider the cylindrical geometry. The correction factor is then the difference between the cylindrical heat source solution and the infinite line source solution, as proposed by Li et al. (2014) as

g(t) = g_FLS + (g_CHS - g_ILS)

Implementation

The calculation of the g-function is separated into two regions: the short-time region and the long-time region. In the short-time region, corresponding to times t < 1 hour, heat interaction between boreholes and axial variations of heat injection rate are not considered. The g-function is calculated using only one borehole and one segment. In the long-time region, corresponding to times t > 1 hour, all boreholes are represented as series of nSeg line segments and the g-function is evaluated as described above.

References

Cimmino, M. and Bernier, M. 2014. A semi-analytical method to generate g-functions for geothermal bore fields. International Journal of Heat and Mass Transfer 70: 641-650.

Cimmino, M. 2018. Fast calculation of the g-functions of geothermal borehole fields using similarities in the evaluation of the finite line source solution. Journal of Building Performance Simulation 11(6): 655-668.

Eskilson, P. 1987. Thermal analysis of heat extraction boreholes. Ph.D. Thesis. Department of Mathematical Physics. University of Lund. Sweden.

Li, M., Li, P., Chan, V. and Lai, A.C.K. 2014. Full-scale temperature response function (G-function) for heat transfer by borehole heat exchangers (GHEs) from sub-hour to decades. Applied Energy 136: 197-205.

Extends from Modelica.Icons.Function (Icon for functions).

Inputs

Type	Name	Default	Description
Integer	nBor		Number of boreholes
Position	cooBor[nBor, 2]		Coordinates of boreholes [m]
Height	hBor		Borehole length [m]
Height	dBor		Borehole buried depth [m]
Radius	rBor		Borehole radius [m]
ThermalDiffusivity	aSoi		Ground thermal diffusivity used in g-function evaluation [m2/s]
Integer	nSeg		Number of line source segments per borehole
Integer	nTimSho		Number of time steps in short time region
Integer	nTimLon		Number of time steps in long time region
Real	ttsMax		Maximum adimensional time for gfunc calculation
Integer	nClu		Number of clusters
Integer	labels[nBor]		Cluster label associated with each data point
Integer	cluSiz[nClu]		Size of the clusters
Real	relTol	0.02	Relative tolerance on distance between boreholes

Outputs

Type	Name	Description
Time	tGFun[nTimSho + nTimLon]	Time of g-function evaluation [s]
Real	g[nTimSho + nTimLon]	g-function

Modelica definition

function gFunction "Evaluate the g-function of a bore field" extends Modelica.Icons.Function; input Integer nBor "Number of boreholes"; input Modelica.Units.SI.Position cooBor[nBor,2] "Coordinates of boreholes"; input Modelica.Units.SI.Height hBor "Borehole length"; input Modelica.Units.SI.Height dBor "Borehole buried depth"; input Modelica.Units.SI.Radius rBor "Borehole radius"; input Modelica.Units.SI.ThermalDiffusivity aSoi "Ground thermal diffusivity used in g-function evaluation"; input Integer nSeg "Number of line source segments per borehole"; input Integer nTimSho "Number of time steps in short time region"; input Integer nTimLon "Number of time steps in long time region"; input Real ttsMax "Maximum adimensional time for gfunc calculation"; input Integer nClu "Number of clusters"; input Integer labels[nBor] "Cluster label associated with each data point"; input Integer cluSiz[nClu] "Size of the clusters"; input Real relTol = 0.02 "Relative tolerance on distance between boreholes"; output Modelica.Units.SI.Time tGFun[nTimSho + nTimLon] "Time of g-function evaluation"; output Real g[nTimSho+nTimLon] "g-function"; protected Modelica.Units.SI.Time ts=hBor^2/(9*aSoi) "Characteristic time"; Modelica.Units.SI.Time tSho_min=1 "Minimum time for short time calculations"; Modelica.Units.SI.Time tSho_max=3600 "Maximum time for short time calculations"; Modelica.Units.SI.Time tLon_min=tSho_max "Minimum time for long time calculations"; Modelica.Units.SI.Time tLon_max=ts*ttsMax "Maximum time for long time calculations"; Modelica.Units.SI.Time tSho[nTimSho] "Time vector for short time calculations"; Modelica.Units.SI.Time tLon[nTimLon] "Time vector for long time calculations"; Integer n_max = max(cluSiz.*cluSiz); Modelica.Units.SI.Distance dis[nClu,nClu,n_max] "Separation distance between boreholes"; Modelica.Units.SI.Distance dis_ij "Separation distance between boreholes"; Integer wDis[nClu,nClu,n_max] "Number of occurence of separation distances"; Integer n_dis[nClu,nClu]; Modelica.Units.SI.Radius rLin=0.0005*hBor "Radius for evaluation of same-borehole line source solutions"; Real hSegRea[nSeg] "Real part of the FLS solution"; Real hSegMir[2*nSeg-1] "Mirror part of the FLS solution"; Modelica.Units.SI.Height dSeg "Buried depth of borehole segment"; Real A[nSeg*nClu+1, nSeg*nClu+1] "Coefficient matrix for system of equations"; Real B[nSeg*nClu+1] "Coefficient vector for system of equations"; Real X[nSeg*nClu+1] "Solution vector for system of equations"; Real FLS "Finite line source solution"; Real ILS "Infinite line source solution"; Real CHS "Cylindrical heat source solution"; Boolean found "Flag, true if a cluster has been found"; algorithm // Distances between borehole clusters n_dis := zeros(nClu,nClu); wDis := zeros(nClu,nClu,n_max); for i in 1:nBor loop for j in i:nBor loop // Distance between boreholes if i <> j then dis_ij := sqrt((cooBor[i,1] - cooBor[j,1])^2 + (cooBor[i,2] - cooBor[j,2])^2); else dis_ij := rLin; end if; found := false; for n in 1:n_dis[labels[i],labels[j]] loop if abs(dis_ij - dis[labels[i],labels[j],n]) / dis[labels[i],labels[j],n] < relTol then wDis[labels[i],labels[j],n] := wDis[labels[i],labels[j],n] + 1; found := true; if i <> j then wDis[labels[j],labels[i],n] := wDis[labels[j],labels[i],n] + 1; end if; break; end if; end for; if not found then n_dis[labels[i],labels[j]] := n_dis[labels[i],labels[j]] + 1; wDis[labels[i],labels[j],n_dis[labels[i],labels[j]]] := wDis[labels[i],labels[j],n_dis[labels[i],labels[j]]] + 1; dis[labels[i],labels[j],n_dis[labels[i],labels[j]]] := dis_ij; if i <> j then n_dis[labels[j],labels[i]] := n_dis[labels[j],labels[i]] + 1; wDis[labels[j],labels[i],n_dis[labels[j],labels[i]]] := wDis[labels[j],labels[i],n_dis[labels[j],labels[i]]] + 1; dis[labels[j],labels[i],n_dis[labels[j],labels[i]]] := dis_ij; end if; end if; end for; end for; // Generate geometrically expanding time vectors tSho := Buildings.Fluid.Geothermal.Borefields.BaseClasses.HeatTransfer.ThermalResponseFactors.timeGeometric( tSho_min, tSho_max, nTimSho); tLon := Buildings.Fluid.Geothermal.Borefields.BaseClasses.HeatTransfer.ThermalResponseFactors.timeGeometric( tLon_min, tLon_max, nTimLon); // Concatenate the short- and long-term parts tGFun := cat(1, {0}, tSho[1:nTimSho - 1], tLon); // ----------------------- // Short time calculations // ----------------------- g[1] := 0.; for k in 1:nTimSho loop // Finite line source solution FLS := Buildings.Fluid.Geothermal.Borefields.BaseClasses.HeatTransfer.ThermalResponseFactors.finiteLineSource( tSho[k], aSoi, rLin, hBor, dBor, hBor, dBor); // Infinite line source solution ILS := 0.5* Buildings.Fluid.Geothermal.Borefields.BaseClasses.HeatTransfer.ThermalResponseFactors.infiniteLineSource( tSho[k], aSoi, rLin); // Cylindrical heat source solution CHS := 2*Modelica.Constants.pi* Buildings.Fluid.Geothermal.Borefields.BaseClasses.HeatTransfer.ThermalResponseFactors.cylindricalHeatSource( tSho[k], aSoi, rBor, rBor); // Correct finite line source solution for cylindrical geometry g[k+1] := FLS + (CHS - ILS); end for; // ---------------------- // Long time calculations // ---------------------- // Initialize coefficient matrix A for m in 1:nClu loop for u in 1:nSeg loop // Tb coefficient in spatial superposition equations A[(m-1)*nSeg+u,nClu*nSeg+1] := -1; // Q coefficient in heat balance equation A[nClu*nSeg+1,(m-1)*nSeg+u] := cluSiz[m]; end for; end for; // Initialize coefficient vector B // The total heat extraction rate is constant B[nClu*nSeg+1] := nBor*nSeg; // Evaluate thermal response matrix at all times for k in 1:nTimLon-1 loop for i in 1:nClu loop for j in i:nClu loop // Evaluate Real and Mirror parts of FLS solution // Real part for m in 1:nSeg loop hSegRea[m] := Buildings.Fluid.Geothermal.Borefields.BaseClasses.HeatTransfer.ThermalResponseFactors.finiteLineSource_Equivalent( tLon[k + 1], aSoi, dis[i,j,1:n_dis[i,j]], wDis[i,j,1:n_dis[i,j]], hBor/nSeg, dBor, hBor/nSeg, dBor + (m - 1)*hBor/nSeg, cluSiz[i], n_dis[i,j], includeMirrorSource=false); end for; // Mirror part for m in 1:(2*nSeg-1) loop hSegMir[m] := Buildings.Fluid.Geothermal.Borefields.BaseClasses.HeatTransfer.ThermalResponseFactors.finiteLineSource_Equivalent( tLon[k + 1], aSoi, dis[i,j,1:n_dis[i,j]], wDis[i,j,1:n_dis[i,j]], hBor/nSeg, dBor, hBor/nSeg, dBor + (m - 1)*hBor/nSeg, cluSiz[i], n_dis[i,j], includeRealSource=false); end for; // Add thermal response factor to coefficient matrix A for u in 1:nSeg loop for v in 1:nSeg loop A[(i-1)*nSeg+u,(j-1)*nSeg+v] := hSegRea[abs(u-v)+1] + hSegMir[u+v-1]; A[(j-1)*nSeg+v,(i-1)*nSeg+u] := (hSegRea[abs(u-v)+1] + hSegMir[u+v-1]) * cluSiz[i] / cluSiz[j]; end for; end for; end for; end for; // Solve the system of equations X := Modelica.Math.Matrices.solve(A,B); // The g-function is equal to the borehole wall temperature g[nTimSho+k+1] := X[nClu*nSeg+1]; end for; // Correct finite line source solution for cylindrical geometry for k in 2:nTimLon loop // Infinite line source ILS := 0.5* Buildings.Fluid.Geothermal.Borefields.BaseClasses.HeatTransfer.ThermalResponseFactors.infiniteLineSource( tLon[k], aSoi, rLin); // Cylindrical heat source CHS := 2*Modelica.Constants.pi* Buildings.Fluid.Geothermal.Borefields.BaseClasses.HeatTransfer.ThermalResponseFactors.cylindricalHeatSource( tLon[k], aSoi, rBor, rBor); g[nTimSho+k] := g[nTimSho+k] + (CHS - ILS); end for; end gFunction;

Buildings.Fluid.Geothermal.Borefields.BaseClasses.HeatTransfer.ThermalResponseFactors.infiniteLineSource

Infinite line source model for borehole heat exchangers

Information

This function evaluates the infinite line source solution. This solution gives the relation between the constant heat transfer rate (per unit length) injected by a line heat source of infinite length and the temperature raise in the medium. The infinite line source solution is defined by

where ΔT(t,r) is the temperature raise after a time t of constant heat injection and at a distance r from the line source, Q' is the heat injection rate per unit length, k_s is the soil thermal conductivity and h_ILS is the infinite line source solution.

The infinite line source solution is given by the exponential integral

where α_s is the ground thermal diffusivity. The exponential integral is implemented in Buildings.Utilities.Math.Functions.exponentialIntegralE1.

Extends from Modelica.Icons.Function (Icon for functions).

Inputs

Type	Name	Description
Real	t	Time
Real	aSoi	Ground thermal diffusivity
Real	dis	Radial distance between borehole axes

Outputs

Type	Name	Description
Real	h_ils	Thermal response factor of borehole 1 on borehole 2

Modelica definition

function infiniteLineSource "Infinite line source model for borehole heat exchangers" extends Modelica.Icons.Function; input Real t "Time"; input Real aSoi "Ground thermal diffusivity"; input Real dis "Radial distance between borehole axes"; output Real h_ils "Thermal response factor of borehole 1 on borehole 2"; algorithm h_ils := if t > 0.0 then Buildings.Utilities.Math.Functions.exponentialIntegralE1(dis^2/(4*aSoi*t)) else 0.0; end infiniteLineSource;

Buildings.Fluid.Geothermal.Borefields.BaseClasses.HeatTransfer.ThermalResponseFactors.shaGFunction

Returns a SHA1 encryption of the formatted arguments for the g-function generation

Information

This function returns the SHA1 encryption of its arguments.

Implementation

Each argument is formatted in exponential notation with four significant digits, for example 1.234e+001, with no spaces or other separating characters between each argument value. To prevent too long strings that can cause buffer overflows, the sha encoding of each argument is computed and added to the next string that is parsed.

The SHA1 encryption is computed using Buildings.Utilities.Cryptographics.sha.

Extends from Modelica.Icons.Function (Icon for functions).

Inputs

Type	Name	Description
Integer	nBor	Number of boreholes
Position	cooBor[nBor, 2]	Coordinates of boreholes [m]
Height	hBor	Borehole length [m]
Height	dBor	Borehole buried depth [m]
Radius	rBor	Borehole radius [m]
ThermalDiffusivity	aSoi	Ground thermal diffusivity used in g-function evaluation [m2/s]
Integer	nSeg	Number of line source segments per borehole
Integer	nClu	Number of clusters for g-function calculation
Integer	nTimSho	Number of time steps in short time region
Integer	nTimLon	Number of time steps in long time region
Real	ttsMax	Maximum adimensional time for gfunc calculation

Outputs

Type	Name	Description
String	sha	SHA1 encryption of the g-function arguments

Modelica definition

function shaGFunction "Returns a SHA1 encryption of the formatted arguments for the g-function generation" extends Modelica.Icons.Function; input Integer nBor "Number of boreholes"; input Modelica.Units.SI.Position cooBor[nBor,2] "Coordinates of boreholes"; input Modelica.Units.SI.Height hBor "Borehole length"; input Modelica.Units.SI.Height dBor "Borehole buried depth"; input Modelica.Units.SI.Radius rBor "Borehole radius"; input Modelica.Units.SI.ThermalDiffusivity aSoi "Ground thermal diffusivity used in g-function evaluation"; input Integer nSeg "Number of line source segments per borehole"; input Integer nClu "Number of clusters for g-function calculation"; input Integer nTimSho "Number of time steps in short time region"; input Integer nTimLon "Number of time steps in long time region"; input Real ttsMax "Maximum adimensional time for gfunc calculation"; output String sha "SHA1 encryption of the g-function arguments"; protected constant String formatStrGen = "1.3e" "String format for general parameters"; constant String formatStrCoo = ".2f" "String format for coordinate"; algorithm sha := Buildings.Utilities.Cryptographics.sha(String(nBor, format=formatStrGen)); sha := Buildings.Utilities.Cryptographics.sha(sha + String(hBor, format=formatStrGen)); sha := Buildings.Utilities.Cryptographics.sha(sha + String(dBor, format=formatStrGen)); sha := Buildings.Utilities.Cryptographics.sha(sha + String(rBor, format=formatStrGen)); sha := Buildings.Utilities.Cryptographics.sha(sha + String(aSoi, format=formatStrGen)); sha := Buildings.Utilities.Cryptographics.sha(sha + String(nSeg, format=formatStrGen)); sha := Buildings.Utilities.Cryptographics.sha(sha + String(nClu, format=formatStrGen)); sha := Buildings.Utilities.Cryptographics.sha(sha + String(nTimSho, format=formatStrGen)); sha := Buildings.Utilities.Cryptographics.sha(sha + String(nTimLon, format=formatStrGen)); sha := Buildings.Utilities.Cryptographics.sha(sha + String(ttsMax, format=formatStrGen)); for i in 1:nBor loop sha := Buildings.Utilities.Cryptographics.sha(sha + String(cooBor[i, 1], format=formatStrCoo)); sha := Buildings.Utilities.Cryptographics.sha(sha + String(cooBor[i, 2], format=formatStrCoo)); end for; end shaGFunction;

Buildings.Fluid.Geothermal.Borefields.BaseClasses.HeatTransfer.ThermalResponseFactors.timeGeometric

Geometric expansion of time steps

Information

This function attemps to build a vector of length nTim with a geometric expansion of the time variable between dt and t_max.

If t_max > nTim*dt, then a geometrically expanding vector is built as

t = [dt, dt*(1-r²)/(1-r), ... , dt*(1-rⁿ)/(1-r), ... , t_max],

where r is the geometric expansion factor.

If t_max < nTim*dt, then a linearly expanding vector is built as

t = [dt, 2*dt, ... , n*dt, ... , nTim*dt]

Extends from Modelica.Icons.Function (Icon for functions).

Inputs

Type	Name	Description
Duration	dt	Minimum time step [s]
Time	t_max	Maximum value of time [s]
Integer	nTim	Number of time values

Outputs

Type	Name	Description
Real	t[nTim]	Time vector

Modelica definition

function timeGeometric "Geometric expansion of time steps" extends Modelica.Icons.Function; input Modelica.Units.SI.Duration dt "Minimum time step"; input Modelica.Units.SI.Time t_max "Maximum value of time"; input Integer nTim "Number of time values"; output Real t[nTim] "Time vector"; protected Real r(min=1.) "Expansion rate of time values"; Real dr "Error on expansion rate evaluation"; algorithm if t_max > nTim*dt then // Determine expansion rate (r) dr := 1e99; r := 2; while abs(dr) > 1e-10 loop dr := (1+t_max/dt*(r-1))^(1/nTim) - r; r := r + dr; end while; // Assign time values for i in 1:nTim-1 loop t[i] := dt*(1-r^i)/(1-r); end for; t[nTim] := t_max; else // Number of time values too large for chosen parameters: // Use a constant time step for i in 1:nTim loop t[i] := i*dt; end for; end if; end timeGeometric;