.Buildings.Fluid.Geothermal.ZonedBorefields.BaseClasses.HeatTransfer.GroundTemperatureResponse .Buildings.Fluid.Geothermal.ZonedBorefields.BaseClasses.HeatTransfer.GroundTemperatureResponseThis 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
rcel (rcel in this model) which follows the
exponential growth :
where nCel is the number of consecutive cells which can have the same size.
Decreasing rcel 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 hIJ,uv is the thermal response factor of segment
v of borehole J onto segment u of a
borehole I, ks 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 Tb,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. tk
is the last discrete aggregation time step, meaning that the current time t
satisfies tk≤t≤tk+1.
Δtagg(=tk+1-tk) 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 ΔTb,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 tk+1, this term is assumed to have a linear time derivative, which is given by the difference between ΔTb,I,u*(tk+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, ΔTb,I,u(tk).
The second term ΔTb,q,I,u(t) concerns the ongoing aggregation time step.
To obtain the time derivative of this term, the thermal response factor hII,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 QI,u at
the borehole wall.
With the two terms in the expression of ΔTb,I,u(t) expressed
as time derivatives, ΔTb,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.
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.