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
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_Erfint	Integral of the error function
finiteLineSource_Integrand	Integrand function for finite line source evaluation
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.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).

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Modelica definition

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).

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Modelica definition

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).

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Modelica definition

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).

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Modelica definition

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

Integrand function for finite line source evaluation

Information

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

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

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Modelica definition

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), and Cimmino (2018)) 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, each borehole is 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. DOI: 10.1080/19401493.2017.1423390.

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).

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Modelica definition

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).

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Modelica definition

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).

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Modelica definition

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).

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Modelica definition