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A Multiple Load Aggregation Algorithm for Annual
Hourly Simulations of GCHP Systems
a
b
c
Michel A. Bernier , Patrice Pinel , Richard Labib & Raphaël Paillot
d
a
Département de génie mécanique
b
Buildings Group, CANMET Energy Technology Centre, Ottawa, Ontario, Canada
c
Département de mathématiques et génie industriel, École Polytechnique de Montréal,
Quebec, Canada
d
Département de génie civil et urbanisme, INSA, Lyon, France
Published online: 28 Feb 2011.
To cite this article: Michel A. Bernier , Patrice Pinel , Richard Labib & Raphal Paillot (2004) A Multiple Load Aggregation
Algorithm for Annual Hourly Simulations of GCHP Systems, HVAC&R Research, 10:4, 471-487
To link to this article: http://dx.doi.org/10.1080/10789669.2004.10391115
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VOLUME 10, NUMBER 4
HVAC&R RESEARCH
OCTOBER 2004
A Multiple Load Aggregation Algorithm for
Annual Hourly Simulations of GCHP Systems
Michel A. Bernier, Ph.D.
Patrice Pinel
Member ASHRAE
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Richard Labib
Raphaël Paillot
Received January 17, 2003; accepted January 21, 2004
This article presents a technique to aggregate heating/cooling loads when using the cylindrical
heat source method (CHS) to perform annual hourly energy simulations of ground-coupled heat
pump (GCHP) systems. The technique, referred to as “multiple load aggregation algorithm” (or
MLAA), uses two major thermal history periods, referred to as “past” and “immediate.” In
addition, the MLAA accounts for thermal interference among boreholes by numerically solving
the two-dimensional temperature field in the borefield. Results of a comparison between the
MLAA and the duct storage (DST) model are presented. Several cases are examined with two
different borefields and several load profiles. Results obtained for one- and ten-year simulations
show that the MLAA is in very good agreement with the DST model. In the worst case, the maximum difference in fluid temperature is of the order of 2 K (3.6°F). This level of precision is more
than adequate to perform accurate hourly simulations of GCHP systems.
INTRODUCTION
There are two main factors that establish the economic feasibility of ground-coupled heat
pump (GCHP) systems: the length of the ground heat exchanger and potential energy savings.
The length of ground heat exchangers can be determined using commercially available computer programs (Shonder et al. 2000). Generally, these programs require peak heating and cooling loads as well as the amount of monthly and annual energy rejected into the ground. When
this approach is used, it is possible to estimate—with variable degrees of accuracy—the annual
energy consumption of the system (heat pump[s] and circulating pump[s]) using averaged coefficients of performance (COP) and/or equivalent full-load heating and cooling hours.
In contrast, annual hourly simulations of GCHP systems provide more accurate results for a
number of reasons that are all linked to the fact that the fluid loop temperature is evaluated at
every time step (usually every hour). With precise hourly evaluation of the fluid loop temperature, it is possible to evaluate the COP more accurately. Consequently, the energy consumption
of each heat pump as well as the hourly running time of the circulating pump(s) are more precise. In addition, one can verify how close the fluid loop temperature is to the minimum and
maximum temperatures tolerated by the heat pump(s) and resize the borefield accordingly.
Another important benefit of hourly system simulations is the ability to implement sophisticated
system control and operation strategies.
A number of researchers have used annual hourly simulations in their work. Deerman and
Kavanaugh (1991) used the so-called cylindrical heat source (CHS) to predict heat pump
Michel A. Bernier is a professor in the Département de génie mécanique and Richard Labib is an assistant professor in
the Département de mathématiques et génie industriel, École Polytechnique de Montréal, Quebec, Canada. Patrice Pinel
is with Buildings Group, CANMET Energy Technology Centre, Ottawa, Ontario, Canada. Raphaël Paillot is an undergraduate student in the Département de génie civil et urbanisme, INSA, Lyon, France.
471
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HVAC&R RESEARCH
entering water temperature for several consecutive hours. Thornton et al. (1997) used the duct
storage (DST) model of Hellström (1991) in the TRNSYS environment to perform hourly
simulations of a family housing unit. Yavuzturk and Spitler (1999) extended the g-functions
concept of Eskilson (1987) to short time steps in order to perform hourly simulations. These
techniques can be used for hourly simulations by dividing ground loads into individual heat
pulses that are then superimposed in time for each time step. As noted by Yavuzturk and Spitler
(1999), the number of superposition calculations is proportional to the square of the number of
time steps. Therefore, if care is not taken, annual simulations composed of 8760 hourly time
steps can be computationally intensive.
However, one does not need to carry individual loads throughout the simulation since the
immediate thermal history (say the last few days) is more important than the past thermal history. Recognizing this characteristic, Yavuzturk and Spitler (1999) introduced the concept of
aggregation for loads that are not part of the immediate thermal history. With this technique,
loads are averaged over blocks of time, and the resulting aggregated load is assumed to prevail
over the full period covered by each block. A user-defined minimum hourly history (immediate
history) period sets the number of hours for which loads are not aggregated. Using a minimum
hourly history period of 192 hours and load aggregation periods of 730 hours, Yavuzturk and
Spitler (1999) show no significant differences in the prediction of the exiting temperatures when
compared to a nonaggregated scheme. The simulation time is reduced by 90% and 99% for
1-year and 20-year simulations, respectively.
Bernier (2001) performed annual hourly simulations of GCHP systems using the CHS
method and a simple load aggregation scheme that consisted of splitting the ground loads into
two time periods. The first period covered the immediate thermal history, i.e., the ground loads
of the last few hours preceding the current time. The thermal history prior to the immediate thermal history (past history) is aggregated into a single averaged ground load. One-year simulations were performed with immediate thermal history ranging from 168 to 1344 hours. For these
two periods, the maximum difference (over one year) in the outlet temperature predicted by this
method and the DST model is 2.2 K (4.0°F) and 0.8 K (1.4°F), respectively. Since heat pump
performance changes are not significant over these temperature ranges, annual heat pump(s) and
circulating pump(s) energy consumption can be computed rapidly and accurately even when the
immediate thermal history is short. However, when precise outlet temperatures are required, the
immediate thermal history has to be relatively long, which results in excessively long simulation
times.
Part of the differences between the DST model and the results of Bernier (2001) can also be
attributed to thermal interference between boreholes. This thermal interference is evaluated
using the spatial superposition principle in the DST model, while Bernier (2001) used the concentric cylinder technique proposed by Kavanaugh and Rafferty (1997).
PROBLEM FORMULATION
A schematic representation of a GCHP system coupled to a vertical U-tube ground heat
exchanger is presented in Figure 1. When performing annual simulations, hourly building loads,
identified by the letter L in Figure 1, are usually known. They are either calculated from a separate load calculation program or internally within the GCHP system simulation program (Bernier and Randriamiarinjatovo 2001). The objective of annual hourly simulations is to calculate
the power, identified by the letter P in Figure 1, required by each heat pump and by the circulating pump(s) (Ppump in Figure 1). The success of this calculation hinges on an accurate determination of the fluid temperature leaving the ground, Tout,ground. The fluid temperature leaving the
ground depends on the borefield configuration and soil/grout properties and on the total amount
of energy rejected/absorbed in the ground, q. In turn, this value of q depends on heat
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VOLUME 10, NUMBER 4, OCTOBER 2004
473
Figure 1. Schematic representation of a GCHP system.
Figure 2. Cross section of a vertical borehole. Tg is the undisturbed far-field ground temperature.
rejected/absorbed by each heat pump in the loop (this quantity is denoted by the letter Q in Figure 1). Thus, when performing annual simulations, the objective is to obtain accurate hourly values of Tout,ground for a given history of hourly ground loads (q).
There are several techniques available to obtain Tout,ground, some of which were mentioned in
the introduction. This article follows the approach of Bernier (2001), which used the CHS analytical solution to get Tout,ground. Two improvements to this earlier work are proposed. First, a
multiple load aggregation algorithm (MLAA) is proposed where ground loads are aggregated
over multiple time periods. In addition, the MLAA accounts for thermal interference among
boreholes by numerically solving the two-dimensional temperature field in the borefield.
MULTIPLE-LOAD AGGREGATION ALGORITHM
Theory
As mentioned earlier, the objective of annual ground heat exchanger simulations is to predict
the temperature of the fluid circulating in a series of boreholes exchanging heat with the adjoining ground. Let’s first consider the case of a single vertical borehole as depicted in the cross section of Figure 2. Three temperature levels are indicated in Figure 2: Tf, the average fluid
temperature in the U-tube; Tw, the borehole wall temperature; and Tg, the undisturbed ground
temperature. This borehole is subjected to varying ground loads, q(t).
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HVAC&R RESEARCH
For a constant ground load, q, the borehole wall temperature at time t, TW,t, can be obtained
using the CHS analytical solution (Carslaw and Jaeger 1947).
q G ( Fo )
T W,t = T g – --- --------------L ks
(1)
where
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q
L
G(Fo)
ks
Fo
= heat transfer rate in watts (a positive q value implies heating, i.e., heat transfer from the ground
to the fluid)
= borehole length, m
= CHS analytical solution
= ground thermal conductivity, W/m⋅K
= Fourier number defined as:
Fo = 4αt / 24db2
(2)
where
α
t
db
= ground thermal diffusivity, m2/day
= time, h
= borehole diameter, m
In order to obtain Tf,t , it is customary to neglect the thermal capacitance of the borehole.
Under this steady-state assumption, the fluid temperature is given by
q G ( Fo )
q
T f,t = T g – --- R b – --- --------------- ,
L ks
L
(3)
where Rb is the equivalent steady-state borehole thermal resistance, °C⋅m/W (°F⋅ft⋅h/Btu).
It is important to note that the CHS method accounts for one-dimensional heat transfer (in the
radial direction) between the borehole and the undisturbed far-field. Heat transfer in the axial
and azimuthal directions is neglected. Since heat losses (gains) from the borefield to the ambient
air and to the ground region below the borefield are not accounted for, the cylinder source
approach essentially “collapses” to a cylinder segment source approach.
The MLAA proposed here is merely an extension of Equation 3 for cases where ground loads
vary with time. It is perhaps best explained by referring to the schematic provided in Figure 3,
where two major thermal history periods—referred to as “past” and “immediate”—are identi-
Figure 3. Multiple load aggregation scheme.
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VOLUME 10, NUMBER 4, OCTOBER 2004
475
fied. The immediate thermal history, noted by h in Figure 3, is set to Nh hours. The past thermal
history is subdivided into four time intervals, denoted by the indices d, w, m, and y. The indices
were chosen to reflect the fact that the time periods are of the order of a day (d), a week (w), a
month (m), and years (y).
Constant ground loads are assumed to prevail over a given time interval. For example, qt is
the hourly ground load prevailing during the time period from t-1 to t hour. Ground loads in the
immediate thermal history, from q t – N + 1 to qt, are not aggregated. On the other hand, values of
h
q in the past thermal history are aggregated and are represented by an overbar. For instance, qd,t
is an aggregated load obtained by taking the average of all ground loads of the last day, i.e.,
between the start of the (Ny+Nm+Nw+1)th hour and the end of the (Ny+Nm+Nw+Nd)th hour.
Each of these periods, except the yearly one, has a fixed length Xi. In the beginning of the
simulation, when the current time is smaller than Xh (t < Xh), the non-aggregated period, h, contains a number of hours equal to the time (Nh = t). After that, it contains Xh hours (Nh = Xh). The
daily aggregated period, d, contains 0 hours (Nd = 0) until the time reaches Xh + Xd. After that, it
contains Xd hours (Nd = Xd). The same applies for the weekly and monthly periods (Nw = 0 or
Xw and Nm = 0 or Xm). The yearly period, Ny, does not have a fixed length since it contains the
rest of the hours (Ny = t – Nm – Nw – Nd – Nh). As an example, let’s assume that Xh, Xd, Xw, and
Xm are fixed at 12, 48, 168, and 360 hours. Then, at t=300 hours we get Nh, Nd, Nw, Nm, and Ny
of 12, 48, 168, 0, and 72 hours, respectively.
With reference to the nomenclature presented in Figure 3 and following the approach
described by Bernier (2001), the mean fluid temperature in the ground heat exchanger at a time
t, i.e., Tf,t, is given by
qt Rb 1
T f,t = T g – ----------- – -------- ( q y,t [ A – B ] + q m,t [ B – C ] + q w,t [ C – D ] + q d,t [ D – E ]
L ks L
+ qt – N + 1 [ E – F1 ] + qt – N + 2 [ F1 – F2 ] + … + qt – 1 [ FN – 1 – FN ] + qt [ FN ] ,
h
h
h
h
h
where
(4)
A = G ( Fo t = t ), B = G ( Fo t = t – N ), C = G ( Fo t = t – N – N ), D = G ( Fo t = t – N – N
y
y
m
y
m – Nw
)
E = G ( Fo t = N ), F 1 = G ( Fo t = N – 1 ), F 2 = G ( Fo t = N – 2 ), …, F N = G ( Fo t = 1 )
h
h
h
h
qi,t = mean ground loads on each aggregation period (i = d, w, m, and y), W (Btu/h) and
qj = hourly (j = t – Nh + 1…. t) non-aggregated ground loads, W (Btu/h).
For convenience, Equation 4 can be represented as
qt Rb 1
T f,t = T g – ----------- – -------- ( MLAA ) ,
L
ks L
(5)
where MLAA represents all of the terms of the multiple load aggregation algorithm contained in
Equation 4.
The determination of Rb, the equivalent steady-state borehole resistance, has been the subject
of a number of publications. For details on the calculation of Rb, readers are referred to the
works of Remund (1999) and Hellstrom (1991) for two-pipe and four-pipe systems, respectively.
The G-function (not to be confused with the g-function mentioned earlier with regard to the
work of Eskilson [1987] and Yavuzturk and Spitler [1999]) represents the solution to a relatively complex integral not suited for hourly evaluation in annual simulations (Bernier 2001).
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HVAC&R RESEARCH
Based on a suggestion by Shonder (2001), the approximate analytical expression proposed by
Cooper (1976) is used. For a complete discussion of the G-function, the reader is referred to the
work of Pinel (2003).
Returning to Equation 4, it is important to note that the aggregated ground loads are updated
at each time step. In other words, qd,t is an aggregated ground load that only applies to the evaluation of Tf,t. It is also interesting to note that the expression inside the parentheses represents
the sum of Nh + 4 terms. Under a non-aggregated scheme, a summation of t terms would be
required. For a full description of the computer implementation of this algorithm, the reader is
referred to the work of Pinel (2003).
The value of interest, Tout,ground,t is obtained by applying an energy balance to the borefield.
qt
-,
T out,ground,t = T f,t + ------------·
2m C p
T in,ground,t + T out,ground,t
T f,t = ---------------------------------------------------------------- ,
2
(6)
where m· is the total mass flow rate (kg/s) and Cp is the fluid specific heat (J/kg⋅°C).
To summarize, the MLAA proposed here is composed of five distinct time intervals
regrouped under the “past” and “immediate” thermal histories: Nh consecutive hourly values
(considered here to be the “immediate” thermal history) and daily, weekly, monthly, and yearly
time intervals. The objective now is to determine the “optimum” fixed length (Xi) of each of
these time intervals in order to obtain accurate values of Tout,ground while maintaining the computational time at a reasonable level.
Optimization of the Aggregation Periods’ Lengths.
This optimization is done by comparing results from simulations performed using different
aggregation periods to results from simulations with a very long immediate thermal history
composed of 4900 non-aggregated hours. A single borehole is considered for this analysis.
However, the optimized aggregation period lengths are equally applicable to borefields since the
aggregation process is independent of borehole thermal interference calculations.
All characteristics of this borehole pertinent to the simulations are given in Table 1. Two
ground load profiles are used. They are described in Figures 4 and 5. The real asymmetric profile (Bernier 2000), shown on Figure 4, was obtained by a TRNSYS simulation of a building
exposed to a certain climate. It is a very severe profile in which the cooling load is dominant,
resulting in significant heat accumulation in the ground. The synthetic symmetric profile shown
in Figure 5 was obtained using the following mathematical function:
y = f ( x;A,B,C ) + ( – 1 )
+ E × ( –1 )
D
floor ⎛ ------------ ( x – B )⎞
⎝ 8760
⎠
D
floor ⎛ ------------ ( x – B )⎞
⎝ 8760
⎠
× abs { f ( x;A,B,C ) }
(7)
Dπ
× ⎛ signum ⎛ cos ⎛ ------------ ( x – F )⎞ + G⎞ ⎞
⎝
⎝ ⎝ 4380
⎠
⎠⎠
where
3
π
π
168 – C
1
Cπi
πi
f ( x;A,B,C ) = Asin ⎛ ------ ( x – B )⎞ sin ⎛ ------------ ( x – B )⎞ ⎛ -------------------⎞ + ∑ ----- ⎛ cos ⎛ ---------⎞ – 1⎞ × ⎛ sin ------ ( x – B )⎞
⎝ 12
⎠ ⎝ 4380
⎠ ⎝ 168 ⎠
⎠ ⎝ 84
⎠
πi ⎝ ⎝ 84 ⎠
i=1
(8)
In the above equations, y is the load, angles are measured in radians, x is the time variable,
floor is the largest integer less than or equal to the number considered, abs denotes the absolute
value of the expression, and signum is equal to plus or minus one according to the sign of the
VOLUME 10, NUMBER 4, OCTOBER 2004
477
Table 1. Parameters Used in the Validation of the MLAA
Parameter
Values used for validation
-single borehole-
Values used for validation
-borefield-
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Borefield
Number of boreholes
1
36 (6 × 6)
Total borehole length, L
100 m (328 ft)
3600 m (11808 ft)
Undisturbed ground temp., Tg
10°C (50°F)
10°C (50°F)
Distance between boreholes
N/A
8 m (26.4 ft)
Borehole diameter
15.2 cm (6 in.)
15.2 cm (6 in.)
U-tube outer diameter
3.35 cm (1.32 in.)
3.35 cm (1.32 in.)
U-tube inner diameter
2.74 cm (1.08 in.)
2.74 cm (1.08 in.)
U-tube center-to-center half distance
3.10 cm (1.22 in.)
3.10 cm (1.22 in.)
Pipe thermal conductivity, kp
0.42 W/m-K (0.24 Btu/h⋅ft⋅°F)
0.42 W/m⋅K (0.24 Btu/h⋅ft⋅°F)
Calculated borehole thermal resistance, 0.1028 m-K/W
Rb
(0.178 h⋅ft⋅°F/ Btu)
0.1028 m⋅K/W
(0.178 h⋅ft⋅°F/ Btu)
Ground
Grout thermal conductivity, kg
2.6 W/m⋅K (1.5 Btu/h⋅ft⋅°F)
2.6 W/m⋅K (1.5 Btu/h⋅ft⋅°F)
Thermal conductivity of the ground, ks 1.3 W/m⋅K (0.75 Btu/h⋅ft⋅°F)
1.3 W/m⋅K (0.75 Btu/h⋅ft⋅°F)
Thermal diffusivity of the ground, αs
0.0561 m2/day (0.60 ft2/day)
0.0561 m2/day (0.60 ft2/day)
Fluid mass flow rate per borehole, m·
0.1744 kg/s (1384 lbm/h)
0.1744 kg/s (1384 lbm/h)
Fluid specific heat, Cp
3.96 kJ/kg⋅K (0.95 Btu/(lbm⋅°F) 3.96 kJ/kg⋅K (0.95 Btu/(lbm⋅°F)
Fluid density
1022 kg/m3 (63.8 lbm/ft3)
1022 kg/m3 (63.8 lbm/ft3)
Fluid thermal conductivity
0.49 W/m⋅K (0.28 Btu/h⋅ft⋅°F)
0.49 W/m-K (0.28 Btu/h⋅ft⋅°F)
Storage volume
12800 m3 (4.5 × 105 ft3)
230000 m³ (8.27 × 106 ft3)
Thermal conductivity of the ground
layers above and below the borehole
0.04 W/m-K (0.023 Btu/h⋅ft⋅°F) 0.04 W/m⋅K (0.023 Btu/h⋅ft⋅°F)
Thickness of the ground layers above
and below the borehole
1 and 1000 m
(3.28 ft and 3280 ft)
Fluid
DST
1 and 1000 m
(3.28 ft and 3280 ft)
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HVAC&R RESEARCH
Figure 4. Real asymmetric ground loads profile. (Note: heat rejection to the ground is
negative.)
Figure 5. Synthetic symmetric ground loads profile. (Note: heat rejection to the ground is
negative.)
expression evaluated. This synthetic symmetric profile, shown on Figure 5, was obtained using
the following parameters: A = 2000, B = 2190, C = 80, D = 2, E = 0.01, F = 0, G = 0.95.
Pinel (2003) provides a complete description of the use of this equation. The synthetic load
profile generates balanced cooling and heating ground loads. Therefore, it is a milder profile
than the real asymmetric one. Finally, two simulation durations—one and ten years—are used in
order to assess the influence of that variable.
Table 2 shows the maximum difference (over the simulation duration) in fluid temperatures
between simulations using different aggregation periods and the “reference simulation” (the one
using a 4900 hours immediate thermal history). Fifteen different aggregation schemes are used
for each of the four cases (two ground loads profiles and two simulation durations). The first
part of Table 2 and Figure 6 show this maximum difference for fixed periods of 24, 168, and 720
hours for Xd, Xw, and Xm and varying Xh. Each curve on Figure 6 exhibits a similar pattern with
a steep slope for small values of Xh, followed by a decreasing slope for Xh greater than 12 hours
and a somewhat constant value for Xh > 96 hours. In all cases, the maximum difference
decreases by a value less than 0.5°C (0.9°F) when Xh is increased from 12 to 96, while the corresponding computational time increases by approximately 20%. Since one of our goals is to minimize computational time, an immediate thermal history of 12 hours is considered adequate. The
VOLUME 10, NUMBER 4, OCTOBER 2004
479
Table 2. Maximum Difference in Fluid Temperatures between
Simulations Using Different Aggregation Schemes and the Reference Simulations
X
Max Difference
(°C)
Real asymmetric
1 year
Max Difference
(°C)
Real asymmetric
10 years
Max Difference
(°C)
Synthetic symmetric
1 year
Max Difference
(°C)
Synthetic symmetric
10 years
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First part: Simulations Xh-Xd-Xw-Xm = Xh-24-168-720
Xh = 2
1.12
1.57
1.86
1.86
Xh = 6
0.92
1.35
1.30
1.30
Xh = 12
0.84
1.18
1.03
1.03
Xh = 24
0.76
1.09
0.85
0.85
Xh = 48
0.63
1.08
0.70
0.70
Xh = 96
0.56
1.01
0.58
0.58
Second part: Simulations Xh-Xd-Xw-Xm = 12-Xd-168-720
Xd = 24
0.84
1.18
1.03
1.03
Xd = 48
0.79
1.13
0.77
0.77
Xd = 96
1.04
1.28
1.01
1.01
Third part: Simulations Xh-Xd-Xw-Xm = 12-48-Xw-720
Xw = 84
0.92
1.19
1.02
1.02
Xw = 168
0.79
1.13
0.77
0.77
Xw = 336
1.28
1.28
0.97
0.97
Fourth part: Simulations Xh-Xd-Xw-Xm = 12-48-168-Xm
Xm = 360
1.02
1.49
0.68
0.68
Xm = 720
0.79
1.13
0.77
0.77
Xm = 1440
1.17
1.17
0.80
0.80
Figure 6. Maximum temperature difference between the aggregated schemes and the reference simulations as a function of Xh for Xd, Xw, Xm = 24,168,720.
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HVAC&R RESEARCH
second part of the table shows results for fixed Xh, Xw, and Xm of 12, 168, and 720 hours and
varying Xd. These results clearly show that a daily period of 48 hours (Xd = 48) results in temperatures closer to the reference simulation. Similarly, the third part of the table shows better
agreement for a weekly period of 168 hours (Xw = 168). The fourth part of the table shows that a
monthly period of 720 hours gives better results for the real asymmetric load profile, and a
period of 360 hours results in better agreement with the reference simulation for the synthetic
symmetric profile. Once again, since our goal is to keep as few variables as possible in memory,
a monthly period of 360 hours is considered (Xm=360).
According to the second, third, and fourth parts of Table 2, the maximum temperature differences do not decrease monotonically as Xm, Xw, and Xd become smaller. This unexpected behavior can be explained by the fact that decreasing the number of loads in a period (Xm for
example), while improving the quality of the aggregated load in this period (qmt), also increases
the number of time steps located in other coarser periods (Xy in our example), resulting in more
time steps being modeled coarsely. So, there is an optimum, a compromise between keeping a
smaller number of loads with a finer representation (in Xm) or keeping a larger number of loads
with a representation that is not as fine but still finer than the representation in the next aggregation period (Xy).
As shown in the third and fourth columns of Table 2, the maximum temperature differences
are identical for the one-year and the ten-year simulations when the synthetic loads profile is
used. This is simply because this profile is symmetric with equal heat addition and rejection to
the ground on an annual basis. Therefore, the net thermal interference is zero at the end of each
year, which explains why the maximum temperature differences are the same year after year.
Based on the foregoing analysis, periods Xm, Xw, Xd, and Xh of 360, 168, 48, and 12 hours are
recommended for simulations performed using the MLAA described here. It is not claimed that
this analysis is exhaustive and that these periods give optimum results for all conditions encountered in geothermal systems. This analysis mainly serves to show that the recommended periods
provide good results and were not chosen arbitrarily.
THERMAL INTERACTION AMONG BOREHOLES
In most borefields, boreholes interact thermally with each other. Consider, for instance, the
center borehole of a 3 × 3 square borefield shown in Figure 7. Over time, heat diffusion from the
eight neighboring boreholes interacts with heat diffused from the center borehole. The heat from
Figure 7. Schematic representation of a 3 × 3 borefield surrounded by a thermal reservoir (right). Node structure near a borehole (left).
VOLUME 10, NUMBER 4, OCTOBER 2004
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the center borehole is trapped and stored in the ground surrounding the borehole. Consequently,
the ground temperature near the center borehole increases. The same phenomenon occurs for the
other eight boreholes. However, the problem is less severe at the periphery where heat can diffuse unobstructed toward the far-field since there are no neighboring boreholes in that direction.
Following an approach similar to the one used by Kavanaugh and Rafferty (1997), thermal
interaction is accounted for by using a temperature penalty, Tp. This penalty corrects the borehole wall temperature obtained using the CHS solution (Equation 1) by adding the thermal interaction with the other boreholes. It should be stressed that Tp does not represent the ground
temperature increase. Equation 1 can be rewritten as
q G ( Fo )
T W,t = T g – --- --------------- + T p,t ,
L ks
(9)
where the first two terms on the right-hand side evaluate the increase in borehole wall temperature for one borehole, while the third term, Tp,t, accounts for the thermal influence of the other
boreholes. Equation 5 can also be expanded to include the thermal interaction among boreholes.
qt Rb 1
T f,t = T g – ----------- – -------- ( MLAA ) + T p,t
ks L
L
(10)
where Tp,t is the temperature penalty at time t.
In the approach proposed here, the value of Tp,t is obtained by numerically solving the
two-dimensional heat transfer process occurring in the borefield. The numerical approach,
which uses the finite-volume method of Patankar (1980), is described elsewhere (Pinel 2003) so
only the salient features are presented here. The right portion of Figure 7 illustrates the situation
for a 3 × 3 borefield. As shown on this figure, the calculation domain is composed of the
borefield and an adjoining thermal reservoir. A two-dimensional nonuniform Cartesian grid is
used (not shown on Figure 7) with a greater mesh concentration near the boreholes where
thermal gradients are steeper. As illustrated on the left part of Figure 7, each borehole consists of
one square control volume whose surface corresponds to the borehole surface. This was shown
by Pinel (2003) to be a correct way to represent boreholes.
The boundary condition is one of constant far-field temperature on all four sides of the calculation domain. This boundary condition is imposed at a distance of 0.5 km (L in Figure 7) past
the borefield. Based on preliminary simulations with various borefield configurations and several heat rejection rates, this distance is sufficiently far from the borefield that heat diffusion
does not affect the ground temperature at that location.
Following the approach of Patankar (1980), integration of the governing energy equation over
a control volume around a point c in the calculation domain gives a discretized equation of the
form,
ac Tc = aw TW + ae TE + as TS + an TN + b ,
(11)
where TW, TE, TS, and TN refer to temperatures at neighboring points shown on the left part of
Figure 7. The coefficients ac, aw, ae, as, and an account for heat conduction and thermal capacitance. Heat injection is introduced using a source term (the b term in Equation 11) only in grid
points that model the boreholes. By applying this equation to each point of the calculation
domain, a linear set of algebraic equations is obtained. This set is solved using a tridiagonal
matrix algorithm (TDMA).
The evaluation of Tp,t involves a two-step calculation. First, the whole borefield is included in
the calculation domain. An average source term for each borehole is then calculated based on
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the cumulative average ground load. This value, not to be confused with aggregated ground
loads of the MLAA, is obtained by averaging the ground loads from the time of the last calculation of Tp,t to the present time of calculation. Once the calculations are converged, the temperatures of the borehole walls are approximated by linearly interpolating the temperature prevailing
at the four faces (identified by the letters n, e, s, and w in Figure 7) of the control volume representing each borehole. These four temperatures are then averaged to obtain the mean borehole
wall temperature, Tbw. In equation form, this translates into
1 T n,x ∆y c + T c,x ∆y n T e,x ∆x c + T c,x ∆x e T s,x ∆y c + T c,x ∆y s T w,x ∆x c + T c,x ∆x w
T bw,x = --- -------------------------------------------- + ------------------------------------------- + ------------------------------------------- + --------------------------------------------- ,
4
( ∆y c + ∆y n )
( ∆x c + ∆x e )
( ∆y c + ∆y s )
( ∆x c + ∆x w )
(12)
where the x index refers to a specific borehole.
Finally, the average borehole wall temperatures of the entire borefield, T bw,nb , is calculated
using
nb
T bw,nb =
∑ T bw,x
x----------------------=1
nb
,
(13)
where nb is the number of boreholes.
The second step in the evaluation of Tp,t involves the calculation of the borehole wall temperature for a calculation domain composed of a single borehole with the same source term that was
used for the calculation involving the entire borefield. When calculations are converged, Equation 12 is applied to the results to obtain the mean borehole wall temperature from a single borehole, T bw,1b .
T bw,1b = T bw,x
(14)
In summary, T bw,nb is the temperature at the borehole wall when interference is present
(many boreholes) and T bw,1b is the temperature without interference (single borehole). The difference between the two can be considered to be the average effect of the interference at the
walls of all boreholes. The value of Tp,t is obtained by subtracting the wall temperature of a single borehole from the average wall temperature obtained for the entire borefield.
T p,t = T bw,nb – T bw,1b
(15)
Since these calculations require a relatively long computational time, Tp,t is not computed at
every time step. This does not reduce accuracy much since heat transfer in the ground is a slow
process and changes in conditions at one borehole take time to influence conditions at another
borehole. Pinel (2003), using a sensitivity analysis and comparison to results from the DST
model, concluded that computing the thermal interference every two weeks (336 hours) had a
very small effect on accuracy while reducing computation time considerably.
VALIDATION
The MLAA is validated by comparing its results to the DST model operating in the TRNSYS
environment (Hellström et al. 1996). The DST model is considered by many as the most accurate model to predict ground heat exchanger thermal behavior. For example, it was used by
Shonder et al. (2000) as the benchmark for a comparative study on commercial sizing software.
VOLUME 10, NUMBER 4, OCTOBER 2004
483
In the DST model used for the present study, boreholes are positioned (by the DST program
itself) in a cylindrical storage volume. The DST model can simulate top and bottom heat losses,
whereas the MLAA cannot.
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Single Borehole
In order to remove borehole thermal interference from the comparison, a single borehole is
examined first. Figure 8 shows the results of a comparison between the DST model and the
MLAA. These results show the difference between the temperatures at the outlet of the borefield, Tout,ground, predicted by the DST model and the MLAA for the 10th year of a 10-year simulation using the real asymmetric ground load profile. Two different aggregation schemes are
used: the Xh, Xd, Xw, Xm = 12, 48, 168, 360 hours proposed above and the reference simulation
(the one using a very long non-aggregated immediate thermal history, Xh = 4900 hours). Results
show good agreement between the two models. The maximum difference is –1.94 K (–3.49°F)
and –1.05 K (–1.89°F) for the aggregated scheme and the reference simulation, respectively.
The RMS value of the difference over the ten-year period is 0.84 K (1.51°F) and 0.41 K (0.74°F)
for the proposed aggregation scheme and the reference simulation, respectively.
Figure 9 presents the same results for a two-day period during the tenth year. As was shown
with regard to an earlier version of the MLAA (Bernier 2001), the MLAA proposed here follows
the valleys and the peaks of the DST model both for the proposed scheme and for Xh = 4900.
Borefield
In this second validation test, a 6 × 6 borefield is used. The real asymmetric ground load profile described earlier is also used. This profile is extremely severe, as the annual heat addition to
the borefield is in the order of 82 kW. Other pertinent conditions are summarized in Table 1.
The aggregation scheme used is the one proposed above (Xh, Xd, Xw, Xm = 12, 48, 168, 360).
Results are shown in Figure 10, where it can be seen that the two algorithms (MLAA and
DST) are in very good agreement. Effectively, as shown on the top graph, the two algorithms
show a maximum difference of 2.12 K (3.82°F) and an RMS difference of 0.78 K (1.40°F).
Since these differences are very close to the ones calculated for one borehole (1.94 K [3.49°F]
max and 0.84 K [1.51°F] RMS), it can be concluded that the thermal interference calculations
Figure 8. Results for one borehole subjected to the real asymmetric ground loads profile
(last year of a ten-year simulation).
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Figure 9. Results for one borehole subjected to the real asymmetric load profile (48
hours).
Figure 10. Results for 36 boreholes subjected to the real asymmetric load profile.
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VOLUME 10, NUMBER 4, OCTOBER 2004
485
are in good agreement for the two algorithms. The bottom graph of Figure 10 shows the variation of the fluid temperature over the simulation period for both the MLAA and the DST models
(both curves are almost indistinguishable). As shown on this figure, the fluid reaches temperatures hotter than 60°C (140°F) for this 36 borehole simulation. Since maximum fluid temperatures around 33°C (91°F) were encountered for the single borehole subjected to the same
conditions, we can conclude that thermal interference increases the temperature at the wall of
the boreholes by more than 27 K (48.6°F) at some time during this simulation.
It must be stressed that this is very far from a realistic case. This is an exaggerated case
designed to serve as a worst case scenario to thoroughly test thermal interference among boreholes. The authors do not claim that it could be viable to cool a building using a fluid at a temperature over 60°C (140°F), as shown in the figure. The ground temperature calculated by the
DST model at the end of the ten-year period is 44.19°C (111.54°F), while the MLAA predicts a
value of 44.90°C (112.82°F)—a difference of 0.71 K (1.28°F). This difference is relatively
small for such a high ground temperature. We can expect temperatures to be in even better
agreement under realistic conditions.
SIMULATION TIME
Simulations were performed in a TRNSYS-type environment on a pentium 166 Mhz
equipped with 32 Mb RAM and using Windows 98. In these conditions, simulation times in the
order of three minutes per year simulated were observed for the proposed aggregation scheme
(Xh, Xd, Xw, Xm = 12, 48, 168, 360). For comparison, simulation times in the order of 20-25 minutes per year were observed for the simulations using the Xh = 4900 scheme. The number of
boreholes in the borefield does not affect calculation speed considerably. This can be explained
by the fact that ground temperatures are evaluated only once every two weeks (336 time steps).
CONCLUDING REMARKS
This article presents a technique to aggregate heating/cooling loads when using the cylindrical heat source method (CHS) to perform annual hourly energy simulations of ground-coupled
heat pump (GCHP) systems. The technique, referred to as “multiple load aggregation algorithm”
(or MLAA) uses two major thermal history periods, referred to as “past” and “immediate.” The
immediate thermal history is not aggregated, while the past thermal history is subdivided into
four time intervals, with periods of the order of a day, a week, a month, and years. A brief analysis (optimization) is performed to obtain a fixed duration for each of these periods that provide
accurate results while keeping computation times at a reasonable level. In addition, the MLAA
accounts for thermal interference among boreholes by numerically solving the two-dimensional
temperature field in the borefield. Interference is evaluated for every two weeks of simulation
time and, therefore, does not slow calculations significantly.
Results of a comparison between the MLAA and the DST model are presented. Several cases
are examined with two different borefields and several load profiles. Results obtained for oneand ten-year simulations show that the MLAA is in very good agreement with the DST model.
In the worst case, the maximum difference in fluid temperature is of the order of 2 K (3.6°F) for
an extremely severe case. This level of precision is more than adequate to perform accurate
hourly simulations of GCHP systems.
ACKNOWLEDGMENTS
The authors would like to thank Mr. John Shonder of Oak Ridge National Laboratory, who
supplied the coded version of the Cooper equations. This research was financially supported by
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HVAC&R RESEARCH
the National Sciences and Engineering Research Council of Canada, Électricité de France, and
the CANMET Energy Technology Center (Natural Resources Canada).
NOMENCLATURE
ac,aw,ae,as,an =
qj
b
Q
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Cp
db
d
Fo
G(Fo)
h
kg
kp
ks
L
L
m
m·
MLAA
Ni
P
Ppump
q
q i,t
heat conduction and thermal
capacitance coefficients for the
numerical method, W/m°C
= source term of the numerical
method, W/m
= specific heat of the fluid,
J/kg⋅°C
= borehole diameter, m
= aggregated period of the order
of a day
= Fourier number
= cylindrical heat source analytical solution
= immediate thermal history
(order of hours)
= grout thermal conductivity,
W/m·K
= pipe thermal conductivity,
W/m·K
= ground thermal conductivity,
W/m⋅K
= borehole length, m
= hourly building load, W
= aggregated period of the order
of a month
= total mass flow rate of fluid in
the borefield, kg/s
= Multiple Load Aggregation
Algorithim
= number of hours in aggregation
period i (i = h, d, w, m, or y), h
= power required by each heat
pump in a loop, W
= power required by the circulating pump(s) in a loop, W
= ground load, i.e., heat extracted
from the ground (positive q
implies heating), W
= aggregated ground loads for
various periods i = d,w,m, or y,
W (Btu/h)
Rb
t
Tbw
T bw,nb
TC
Tf
Tg
Tin,ground
Tout,ground
Tpt
Tw
TW,TE,TS,TN
w
Xi
y
α
∆x, ∆y
= hourly (j = t – Nh + 1…. t
hours) non-aggregated ground
load at time j, W
= heat rejected/absorbed by each
heat pump in a loop, W
= equivalent steady-state borehole thermal resistance,
°C-m/W
= time, h
= borehole wall temperature
evaluated numerically, °C
= mean borehole wall temperature in the borefield (n boreholes), °C
= ground temperature at a point
in the grid (c for center), °C
= mean fluid temperature, °C
= undisturbed ground temperature, °C
= fluid temperature at the borefield inlet, °C
= fluid temperature at the borefield outlet (heat pumps inlet),
°C
= temperature penalty due to
thermal interference among
boreholes, °C
= borehole wall temperature
evaluated by the CHS, °C
= ground temperature at the four
neighboring points, point in the
grid, °C
= aggregated period of the order
of a week
= pre-fixed number of hours in
aggregation period i (i = h, d,
w, or m), hours
= aggregated period containing
the rest of the hours (order of
years)
= ground thermal diffusivity,
m2/day
= x and y dimensions of the cartesian grid elements, m
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