Buildings.Fluid.Geothermal.ZonedBorefields.BaseClasses.HeatTransfer

Package with ground heat transfer models

Information

This package contains models and functions to solve ground heat transfer around ground heat exchangers. For more information on the model implementation, see the documentation of Buildings.Fluid.Geothermal.ZonedBorefields.BaseClasses.HeatTransfer.GroundTemperatureResponse.

Extends from Modelica.Icons.VariantsPackage (Icon for package containing variants).

Package Content

Name	Description
GroundTemperatureResponse	Model calculating discrete load aggregation
shaKappa	Returns a SHA1 encryption of the formatted arguments for the thermal response factor generation
shiftAggregationCells	Performs the shifting operation for load aggregation and determines the current cell
temperatureResponseMatrix	Evaluate the thermal response factors of a zoned thermal storage
temporalSuperposition	Performs temporal superposition for the load aggregation procedure
Validation	Validation models for the geothermal heat transfer models

Buildings.Fluid.Geothermal.ZonedBorefields.BaseClasses.HeatTransfer.GroundTemperatureResponse

Model calculating discrete load aggregation

Information

This model calculates the ground temperature response to obtain the temperature at the wall of each borehole segment in a geothermal system where heat is being injected into or extracted from the ground.

A load-aggregation scheme based on that developed by Claesson and Javed (2012) is used to calculate the borehole wall temperature response with the temporal and spatial superpositions of ground thermal loads. In its base form, the load-aggregation scheme uses fixed-length aggregation cells to agglomerate thermal load history together, with more distant cells (denoted with a higher cell and vector index) representing more distant thermal history. The more distant the thermal load, the less impactful it is on the borehole wall temperature change at the current time step. Each cell has an aggregation time associated to it denoted by nu, which corresponds to the simulation time (since the beginning of heat injection or extraction) at which the cell will begin shifting its thermal load to more distant cells. To determine nu, cells have a temporal size r_cel (rcel in this model) which follows the exponential growth :

where n_Cel is the number of consecutive cells which can have the same size. Decreasing r_cel will generally decrease calculation times, at the cost of precision in the temporal superposition. rcel is expressed in multiples of the aggregation time resolution (via the parameter tLoaAgg). Then, nu may be expressed as the sum of all rcel values (multiplied by the aggregation time resolution) up to and including that cell in question.

The weighting factors giving the impact of the thermal load in a cell m) for a segment v of borehole J onto the temperature at the wall of segment u of a borehole I at the current time is obtained from analytical thermal response factors:

where h_IJ,uv is the thermal response factor of segment v of borehole J onto segment u of a borehole I, k_s is the thermal conductivity of the soil and ν refers to the vector nu in this model.

At every aggregation time step, a time event is generated to perform the load aggregation steps. First, the thermal loads are shifted. When shifting between cells of different size, total energy is conserved. This operation is illustred in the figure below by Cimmino (2014).

After the cell-shifting operation is performed, the first aggregation cell has its value set to the average thermal load since the last aggregation step. Temporal superposition is then applied by means of a scalar product between the aggregated thermal loads QAgg_flow and the weighting factors κ. The spatial superposition is applied by summing the contributions of all segments on the total temperature variation.

Due to Modelica's variable time steps, the load aggregation scheme is modified by separating the thermal response between the current aggregation time step and everything preceding it. This is done according to :

where T_b,I,u is the borehole wall temperature at segment u of borehole I, Q'_b,J,v is the ground thermal load per borehole length at segment v of borehole J. t_k is the last discrete aggregation time step, meaning that the current time t satisfies t_k≤t≤t_k+1. Δt_agg(=t_k+1-t_k) is the parameter tLoaAgg in the present model.

Thermal interactions between segments affect the borehole wall temperature much slower than the effect of heat extraction at a segment on the temperature variation at the same segment. Thus, spatial superposition in the second sum can be neglected :

where ΔT_b,I,u*(t) is the borehole wall temperature change at segment u of borehole I due to the thermal history prior to the current aggregation step. At every aggregation time step, spatial and temporal superpositions are used to calculate its discrete value. Assuming no heat injection or extraction until t_k+1, this term is assumed to have a linear time derivative, which is given by the difference between ΔT_b,I,u*(t_k+1) (the temperature change from load history at the next discrete aggregation time step, which is constant over the duration of the ongoing aggregation time step) and the total temperature change at the last aggregation time step, ΔT_b,I,u(t_k).

The second term ΔT_b,q,I,u(t) concerns the ongoing aggregation time step. To obtain the time derivative of this term, the thermal response factor h_II,uu is assumed to vary linearly over the course of an aggregation time step. Therefore, because the ongoing aggregation time step always concerns the first aggregation cell, its derivative can be calculated as kappa[i,i,1], the first value in the kappa array, divided by the aggregation time step Δt. The derivative of the temperature change at the borehole wall is then expressed by its multiplication with the heat flow Q_I,u at the borehole wall.

With the two terms in the expression of ΔT_b,I,u(t) expressed as time derivatives, ΔT_b,I,u(t) can itself also be expressed as its time derivative and implemented as such directly in the Modelica equations block with the der() operator.

References

Cimmino, M. 2014. Développement et validation expérimentale de facteurs de réponse thermique pour champs de puits géothermiques, Ph.D. Thesis, École Polytechnique de Montréal.

Claesson, J. and Javed, S. 2012. A load-aggregation method to calculate extraction temperatures of borehole heat exchangers. ASHRAE Transactions 118(1): 530-539.

Parameters

Connectors

Modelica definition

Buildings.Fluid.Geothermal.ZonedBorefields.BaseClasses.HeatTransfer.shaKappa

Returns a SHA1 encryption of the formatted arguments for the thermal response factor 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

Outputs

Modelica definition

Buildings.Fluid.Geothermal.ZonedBorefields.BaseClasses.HeatTransfer.shiftAggregationCells

Performs the shifting operation for load aggregation and determines the current cell

Information

Performs the shifting operation which propagates the thermal load history towards the more distant aggregation cells, and then sets the current cell's value at 0. Additionally, this function also outputs the last filled load aggregation cell.

This is a vectorized implementation of Buildings.Fluid.Geothermal.Borefields.BaseClasses.HeatTransfer.LoadAggregation.shiftAggregationCells, which applies load aggregation on a vector of aggregated load histories.

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

Inputs

Outputs

Modelica definition

Buildings.Fluid.Geothermal.ZonedBorefields.BaseClasses.HeatTransfer.temperatureResponseMatrix

Evaluate the thermal response factors of a zoned thermal storage

Information

This function evaluates the array of segment-to-segment thermal response factors using the analytical finite line source method of and Prieto and Cimmino (2021). The finite line source solution gives the relation between heat extraction at a segment of an equivalent borehole representing a group of boreholes (or in this model, a borefield zone) and the temperature variation at the wall of another segment of another (or the same) equivalent borehole representing another group of boreholes. The total temperature varition at a borehole segment is given by the temporal and spatial superpositions of the thermal reponse factors:

where T_b,I,u is the borehole wall temperature of segment u of borehole I, T_g,I,u is the undisturbed ground temperature of segment u of borehole I, Q'_J,v is the heat injection rate into the ground through the borehole wall per unit length of segment v of borehole J, k_s is the soil thermal conductivity and h_IJ,uv is the thermal response factor of segment v of borehole J onto segment u of borehole I.

The thermal response factor is constructed from the combination of the finite line source (FLS) solution (see Buildings.Fluid.Geothermal.Borefields.BaseClasses.HeatTransfer.ThermalResponseFactors.finiteLineSource_Equivalent), 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 thermal response factors of a bore field, the bore field is first divided into nZon zones of parallel-connected boreholes. Each sone 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.

Since the finite line source solution is based on line sources of heat, rather than cylinders, the thermal response factors are 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) :

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

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.

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

Outputs

Modelica definition

Buildings.Fluid.Geothermal.ZonedBorefields.BaseClasses.HeatTransfer.temporalSuperposition

Performs temporal superposition for the load aggregation procedure

Information

Performs the temporal and spatial superposition operations to obtain the temperature change at the wall of a borehole segment at the current time step. The spatial superposition of one load aggregation cell is the matrix-vector product of the kappa[:,:,k] step response matrix (at a cell k) and the vector of aggregated loads of all borehole segments in cell k, QAgg_flow[:,k]. The temporal superposition is the sum of the contributions of all load aggregation cells. To avoid unnecessary calculations, the current aggregation cell in the simulation is used to truncate the values from the vectors that are not required.

This is a vectorized implementation of Buildings.Fluid.Geothermal.Borefields.BaseClasses.HeatTransfer.LoadAggregation.temporalSuperposition.

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

Inputs

Outputs

Modelica definition