Buildings.Fluid.Geothermal.Borefields.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.Borefields.BaseClasses.HeatTransfer.GroundTemperatureResponse.

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

Package Content

Name	Description
Cylindrical	Heat conduction in a cylinder using the radial discretization as advised by Eskilson
GroundTemperatureResponse	Model calculating discrete load aggregation
LoadAggregation	Package with functions for load aggregation
ThermalResponseFactors	Models for heat transfer outside boreholes
Validation	Validation models for GroundHeatTransfer

Buildings.Fluid.Geothermal.Borefields.BaseClasses.HeatTransfer.Cylindrical

Heat conduction in a cylinder using the radial discretization as advised by Eskilson

Information

Model for radial heat transfer in a hollow cylinder.

If the heat capacity of the material is non-zero, then this model computes transient heat conduction, i.e., it computes a numerical approximation to the solution of the heat equation

ρ c ( ∂ T(r,t) ⁄ ∂t ) = k ( ∂² T(r,t) ⁄ ∂r² + 1 ⁄ r   ∂ T(r,t) ⁄ ∂r ),

where ρ is the mass density, c is the specific heat capacity per unit mass, T is the temperature at location r and time t and k is the heat conductivity. At the locations r=r_a and r=r_b, the temperature and heat flow rate are equal to the temperature and heat flow rate of the heat ports.

If the heat capacity of the material is set to zero, then steady-state heat flow is computed using

Q = 2 π k (T_a-T_b)⁄ ln(r_a ⁄ r_b),

where r_a is the internal radius, r_b is the external radius, T_a is the temperature at port a and T_b is the temperature at port b.

Implementation

To spatially discretize the heat equation, the construction is divided into compartments with nSta ≥ 1 state variables. The state variables are connected to each other through thermal conductors. There is also a thermal conductor between the surfaces and the outermost state variables. Thus, to obtain the surface temperature, use port_a.T (or port_b.T) and not the variable T[1].

Parameters

Connectors

Modelica definition

Buildings.Fluid.Geothermal.Borefields.BaseClasses.HeatTransfer.GroundTemperatureResponse

Model calculating discrete load aggregation

Information

This model calculates the ground temperature response to obtain the temperature at the borehole wall 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 superposition 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.

To determine the weighting factors, the borefield's temperature step response at the borefield wall is determined as

where g(·) is the borefield's thermal response factor known as the g-function, H is the total length of all boreholes and k_s is the thermal conductivity of the soil. The weighting factors kappa (κ in the equation below) for a given cell i are then expressed as follows.

where ν refers to the vector nu in this model and T_step(ν₀)=0.

At every aggregation time step, a time event is generated to perform the load aggregation steps. First, the thermal load is 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.

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 is the borehole wall temperature, T_g is the undisturbed ground temperature, Q is the ground thermal load per borehole length and h = g/(2 π k_s) is a temperature response factor based on the g-function. 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.

Thus, ΔT_b*(t) is the borehole wall temperature change due to the thermal history prior to the current aggregation step. At every aggregation time step, load aggregation and temporal superposition 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*(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(t).

The second term ΔT_b,q(t) concerns the ongoing aggregation time step. To obtain the time derivative of this term, the thermal response factor h 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 (denoted by the parameter dTStepdt in this model) can be calculated as kappa[1], the first value in the kappa vector, divided by the aggregation time step Δt. The derivative of the temperature change at the borehole wall is then expressed as the multiplication of dTStepdt (which only needs to be calculated once at the start of the simulation) and the heat flow Q at the borehole wall.

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

This load aggregation scheme is validated in Buildings.Fluid.Geothermal.Borefields.BaseClasses.HeatTransfer.Validation.Analytic_20Years.

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