# A fast algorithm for the simulation of GCHP systems.

ABSTRACT

Annual hourly (or shorter) energy simulations are an important part of the design and analysis of ground-coupled heat pump systems. In order to evaluate the fluid temperature in the borehole of a geothermal heat pump system, most of the current models express the heat transfer rate as a sum of step changes in heat transfer rate. The borehole temperature is then computed as a superposition of the different contributions of each time step. The main difference between the different models lies in the way the step response is computed, whether a cylindrical heat source method, a line source method, or a tabulated numerical step response approach is used. Since all these methods are based on a convolution scheme, long time simulations are very time consuming since impulse response is recomputed at each time step. Many load aggregation algorithms have been proposed in order to reduce this computational time. In this paper, we present a new algorithm to evaluate the overall response, which is much faster than the classical convolution scheme.

INTRODUCTION

Ground-coupled heat pump (GCHP) systems are very effective in lowering the energy used to heat and cool residential and commercial buildings. The main factor that limits the growth of such systems is the initial cost of the borehole in the ground. As energy cost will increase, this cost will become relatively lower, and it is believed these systems will be very attractive in the near future. Still, the length of the ground exchanger will always be a very important factor, and a lot of research has been done in the design and analysis of this part of the GCHP system. Most of the models are based on the solution of the impulse response on a heat pulse in the borehole. The difference in the models is mostly in the way the heat conduction problem in the ground is solved and the way the interference problem between boreholes is treated A good survey of the different models is given by Yavuzturk (1999). Without going too much into the details of these methods, we may split these methods into two main approaches: analytical and numerical methods. In both cases some workers analyze only a one-dimensional transient problem in the radial direction, where T(r, t) is sought in the field considered, whereas some models are based on the solution of the axisymmetric problem T(r,z,t). A lot can be said about the validation of both approaches. For example, the radial problem does not give a steady-state solution; it has a logarithm singularity at infinity. One may argue about its validity after a long period of time (Eskilson 1987) if the load is constant. In the case of a symmetric annual load, this long-term effect should, however, not be so important. In any case, many design programs, such as the ones by Kavanaugh (1985) and Bernier (2001), use such a solution and give good results, as mentioned by Shonder et al. (1999).

Analytical methods are also given for the axisymmetric problem by Zeng et al. (2002), but most of them solve the radial problem, and the two major solutions used are the line source solution of Kelvin and the cylindrical heat source method. In both cases, they suffer from the fact that the solution is given in terms of a convolution solution, where each term has to be recomputed at each time step. This is the reason why the computing time is proportional to the square on the time interval. This precludes their use for a short time step (an hour or less) and/or for long period of time (a year or more). In order to reduce the computing time, Yavuzturk and Spitler (Yavuzturk 1999; Yavuzturk and Spitler 1999) proposed the concept of aggregation for loads that were applied a long time before the actual time when the temperature is sought. They define a minimum hourly time history (MHTH) where no aggregation is done and also an aggregation period where a mean load for this aggregation period is computed and used in the simulations. This approach diminishes the simulation time by 90% for a year's simulation. Bernier et al. (2004) also proposed a multiple-load aggregation algorithm (MLAA) in order to cope with the same problem. In this paper, we present a very effective algorithm to solve the same heat conduction problem that is not history-dependent and is very efficient.

PROBLEM FORMULATION

The heat exchange problem in a buried vertical borehole can be formulated with respect to the schematic in Figure 1. If we neglect axial temperature variation, the basic problem is to find the temperature distribution T(r,t) satisfying the heat conduction equation,

[1/[alpha]][[[partial derivative]T]/[[partial derivative]t]] = [[[partial derivative].sup.2]T]/[[partial derivative][r.sup.2]] + [1/r][[[partial derivative]T]/[r[partial derivative]r], (1)

for the domain r > [r.sub.b], t > 0, and the following boundary conditions:

T(r, 0) = [T.sub.o], -k[[[partial derivative]T]/[[partial derivative]r]]|[.sub.r = [r.sub.b]] = [q".sub.b](t) = [[q'.sub.b](t)]/[2[pi][r.sub.b]] (2)

where [q'.sub.b] is the heat flow per unit length [q'.sub.b] is often referred to as the heat entering the borehole; here we keep the normal convention as the heat in the positive radial direction). In the case of step-function [q'.sub.b](t) = [q'.sub.o] u(t), the solution is well known and is given in the classic book of Carslaw and Jaeger (1947).

[FIGURE 1 OMITTED]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

where [~.r] = r/([r.sub.b], [~.t]) = ([alpha]t)/[r.sub.b.sup.2] = Fourier number.

This solution is known as the cylindrical heat source method (CHSM). Those using this solution for the analysis of GCHP systems (Kavanaugh 1985; Bernier 2001) use an analytical approximation of the G-function in their computation with the extension of arbitrary loads and the following expression:

T([~.r], [~.t]) - [T.sub.o] = [1/k][N.summation over (i = 1)]([q.sub.i] - [q.sub.i-1])G([~.r], [[alpha](t - [t.sub.i-1])]/[r.sub.b.sup.2]) (4)

In order to describe our new algorithm, we will solve again the heat conduction problem. Whereas Carslaw and Jaeger (1947) use the Laplace transform technique to present their solution, we will use Green's function formalism here. The solution of the general heat problem is well explained by Ozisik (1993):

T(r, t) = [[integral].sub.[r.sub.b].sup.[infinity]][f.sub.i]([rho])H(r, [rho], t)[rho]d[rho] + [[alpha]/k][[integral].sub.0.sup.t][q".sub.b]([tau])[r.sub.b]H(r, [r.sub.b], t - [tau])d[tau] + [[alpha]/k][[integral].sub.0.sup.t]d[tau][[integral].sub.[r.sub.b].sup.[infinity][[dot].q]([rho], [tau])H(r, [rho], t - [tau])[rho]d[rho] (5)

where [[dot].q] is the volumetric heat source distribution, [f.sub.i](r) is the temperature distribution for t = 0, and H (r, [rho], t - [tau]) is what is known as the Green's function associated to the problem. The symbol G is normally associated with this function, but we will use H here in order not to confuse it with the cylindrical heat source function. Since, in our problem [[dot].q] = 0 and with the change of variable, [~.T] = T - [T.sub.o], we have

[~.T](r, t) = [[alpha]/[2[pi]k]][[integral].sub.0.sup.t][q.sub.b]([tau])H(r, [r.sub.b], t - [tau])d[tau]. (6)

Following Ozisik (1993), we can find the Green's function by solving the associated problem with homogeneous boundary conditions:

[1/[alpha]][[[partial derivative]T]/[[partial derivative]t]] = [[[partial derivative].sup.2]T]/[[partial derivative][r.sup.2]] + [1/r][[[partial derivative]T]/[[partial derivative]r]] (7)

for the domain r > b, t > 0, and the following boundary conditions:

T(r, 0) = [f.sub.i](r), -k[[[partial derivative]T]/[[partial derivative]r]]|[sub.r = b] = 0 (8)

Solving the problem and comparing with the Green's formalism solution,

T(r, t) = [[integral].sub.[r.sub.b].sup.[infinity]][f.sub.i]([rho])H(r, [rho], t)[rho]d[rho], (9)

we can get the associated Green's function by association. This associated problem can be solved by the Weber transform, but we will directly use the solution given by Cole (2000),

H(r, [r.sub.b], t - [tau]) = [1/2[pi]][[integral].sub.0.sup.[infinity]][e.sup.-[alpha][[beta].sup.2](t - [tau])] [[[beta]R(r)R([r.sub.b])]/[[J.sub.1.sup.2]([beta][r.sub.b]) + [Y.sub.1.sup.2]([beta][r.sub.b])]]d[beta], (10)

where

R(r) = [J.sub.o]([beta]r)[Y.sub.1]([beta][r.sub.b]) - [J.sub.1]([beta][r.sub.b])[Y.sub.o]([beta]r). (11)

Inserting this solution into Equation 6, we get

[~.T](r, t) = [[alpha]/[2[pi]k]][[integral].sub.0.sup.t][q'.sub.b]([tau])[[integral].sub.0.sup.[infinity]][e.sup.-[alpha][[beta].sup.2](t - [tau])][[[beta]R(r)R([r.sub.b])]/[[J.sub.1.sup.2]([beta][r.sub.b]) + [Y.sub.1.sup.2]([beta][r.sub.b])]]d[beta]d[tau]. (12)

With Equation 11, we can simplify:

R([r.sub.b]) = [J.sub.o]([beta][r.sub.b])[Y.sub.1]([beta][r.sub.b]) - [J.sub.1]([beta][r.sub.b])[Y.sub.o]([beta][r.sub.b]) = [-2/[[pi][beta][r.sub.b]] (13)

[~.T](r, t) = -[alpha]/[[[pi].sup.2][r.sub.b]k][[integral].sub.0.sup.t][q'.sub.b]([tau])[[integral].sub.0.sup.[infinity]][e.sup.-[alpha][[beta].sup.2](t - [tau])][[R(r)]/[[J.sub.1.sup.2]([beta][r.sub.b]) + [Y.sub.1.sup.2]([beta][r.sub.b])]]d[beta]d[tau] (14)

If we apply a step change of heat, the response is:

[~.T](r, t) = [[-[q'.sub.0][alpha]]/[[[pi].sup.2][r.sub.b]k]] [[integral].sub.0.sup.[infinity]][R(r)/[[J.sub.1.sup.2]([beta][r.sub.b]) + [Y.sub.1.sup.2]([beta][r.sub.b])]]d[beta][[integral].sub.0.sup.t][e.sup.-[alpha][[beta].sup.2](t - [tau])]d[tau] (15)

[~.T](r, t)

= [[q'.sub.o]/k][1/[[pi].sup.2]][[integral].sub.0.sup.[infinity]][[([e.sup.-[z.sup.2][~.t]] - 1)[[J.sub.o]([~.r]z)[Y.sub.1](z) - [J.sub.1](z)[Y.sub.o]([~.r]z]]/[[z.sup.2][[J.sub.1.sup.2](z) + [Y.sub.1.sup.2](z)]])dz

= [[q'.sub.o]/k]G([~.r], [~.t]) (16)

where

[~.r] = r/[r.sub.b], [~.t] = [alpha]t/[r.sub.b.sup.2], and z = [beta][r.sub.b], which is the expected result.

NONHISTORY-DEPENDENT SCHEME

As mentioned in the introduction, the major problem with the convolution scheme is that, at each time step, the weighted G-function has to be recomputed, which results in an algorithm that is proportional to [N.sub.2], where N is the number of time steps. The idea of our new scheme is similar to the one of Greengard and Strain (1990) in the context of the boundary element method (BEM) for transient heat transfer problems. In their case, the problem was even greater, since they used, as is the usual case in BEM, the fundamental solution to Green's function as the kernel of the boundary integrals:

H(x, [xi], t, [tau]) = [1/[4[pi](t - [tau])]][e.sup.-[r.sup.2]/[4[alpha](t - [tau])] (17)

where r = |x - [xi]|.

The difficulty arises from the coupling behavior of the Green's function as the space and time integration variables appear in the exponential term. In order to overcome this problem, they had to use some kind of Fourier expansion of the Green's function. In our particular GCHP problem, the integral solution is already in the form of a degenerate kernel since the space variable and the time variable are uncoupled. This facilitates the scheme.

Let's start with the general solution described in the previous section (Equation 14):

[~.T]([~.r], [~.t]) = [-1/[[[pi].sup.2]k]][~.[integral].sub.0.sup.t]][q'.sub.b]([~.t])[[integral].sub.0.sup.[infinity]][e.sup.-[z.sup.2]([~.t] - [~.[tau]])][[[J.sub.o]([~.r]z)[Y.sub.1](z) - [J.sub.1](z)[Y.sub.o]([~.r]z)]/[[J.sub.1.sup.2](z) + [Y.sub.1.sup.2](z)]]dzd[~.[tau]] (18)

If we interchange the order of integration:

[~.T]([~.r], [~.t]) = [-1/[[[pi].sup.2]k]][[integral].sub.0.sup.[infinity]][[[J.sub.o]([~.r]z)[Y.sub.1](z) - [J.sub.1](z)[Y.sub.o]([~.r]z)]/[[J.sub.1.sup.2](z) + [Y.sub.1.sup.2](z)]][~.[integral].sub.0.sup.t][q'.sub.b]([~.[tau]])[e.sup.-[z.sup.2]([~.t] - [~.[tau]])]d[~.[tau]]dz (19)

In order to simplify the notation, we will make the change of variable:

v([~.r], z) = [[[J.sub.o]([~.r]z)[Y.sub.1](z) - [J.sub.1](z)[Y.sub.o]([~.r]z)]/[[J.sub.1.sup.2](z) + [Y.sub.1.sup.2](z)]] (20)

Let us assume that the improper integrals converge correctly and that we can keep a finite number of terms in the approximation of the integral:

[~.T]([~.r], [~.t]) = [-1/[[[pi].sup.2]k]][N.summation over (n = 1)] [v.sub.n]([~.r], z)[[~.integral].sub.0.sup.t][q'.sub.b]([~.[tau]])[e.sup.-[z.sup.2]([~.t] - [~.[tau]])]d[~.[tau]][DELTA][z.sub.n]

= [1/[[pi].sup.2]k][N.summation over (n = 1)][F.sub.n]([~.t])[DELTA][z.sub.n] (21)

At the next time interval, we can evaluate the new temperature distribution:

[~.T][[~.r], ([~.t] + [DELTA][~.t])]

= [-1/[[pi].sup.2]k][N.summation over (n = 1)][v.sub.n]([~.r], z)[[~.[integral].sub.0.sup.t][q'.sub.b]([~.[tau]])[e.sup.-[z.sup.2]([~.[tau]] + [DELTA][~.t] - [~.[tau]])d[~.[tau]] + [[integral].sub.0.sup.[~.t] + [DELTA][~.t]][q'.sub.b]([~.[tau]])[e.sup.-[z.sup.2]([~.t] + [DELTA][~.t] - [~.[tau]])]d[~.[tau]]][DELTA][z.sub.n] (22)

If we assume again that the heat transfer is piecewise constant in each time interval, we can evaluate the second integral and rearrange the terms:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (23)

Most of the time, the temperature at the borehole radius ([~.r] = 1) is sought. In that particular case, we can simplify this last expression:

v([~.r], z) = [[J.sub.o](z)[Y.sub.1](z) = [J.sub.1](z)[Y.sub.o](z)]/[[J.sub.1.sup.2](z) + [Y.sub.1.sup.2](z)]] = -2/[[pi]z[[J.sub.1.sup.2](z) + [Y.sub.1.sup.2](z)]] (24)

We see that, at each time step, the new coefficient [F.sub.n] is computed using the old ones. We do not have to recompute every one of them. This results in an algorithm of order N as we will see in the next section.

SIMULATION RESULTS

In contrast with the usual convolution algorithm, the nonhistory-dependent algorithm described in the previous section needs the replacement of the improper integral by a finite sum. The final results will, therefore, depend on the number of terms kept and on the finite interval of integration replacing the infinite one. No optimization of the procedure was done, but very good results were obtained with a reasonable number of terms in the summation, and the difference between the exact solution and the new one is often less than the aggregation scheme depending, of course, on the minimum hourly history period used.

For [~.r] = 1, the second integral of Equation 22 is given by

[[2[q'.sub.b]([~.t])]/[[[pi].sup.3]k]][[integral].sub.0.sup.[infinity]][[1 - [e.sup.-[z.sup.2][DELTA][~.t]]]/[z[[J.sub.1.sup.2](z) + [Y.sub.1.sup.2](z)]]dz = [[integral].sub.0.sup.[infinity]]f(z, [DELTA][~.t])dz. (25)

For a one-hour time step, and with diffusivity for earth of [alpha] = 0.0023 [m.sup.3]/h (0.02476 [ft.sup.2]/h) and a 0.152 m (6 in.) diameter borehole, a typical value for the reduced time interval is

[DELTA][~.t] = [[alpha][DELTA]t]/[r.sub.b.sup.2] [approximately equal to] 0.4 (26)

As seen in Figure 2, the function f is small for z > 20. For different values of time step, the function varies more for z < 5, so we used a variable z-step size. The following transformation was used to create the variable mesh:

z = x/[c(1 - x)] (27)

which maps 0 [less than or equal to] x [less than or equal to] 1 to 0 [less than or equal to] z < [infinity], so we chose 0 [less than or equal to] x [less than or equal to] [x.sub.max] with

[x.sub.max] = [c[z.sub.max]]/[1 + [c[z.sub.max]]] (28)

In our tests, we used [z.sub.max] = 30 and c = 0.4.

The algorithm was tested for hourly simulations for different time periods. Two ground load profiles were used for the simulation. They are the same as used by Bernier et al. (2004) and Pinel (2003). They are called synthetic symmetric ground load profile (Figure 3) and synthetic asymmetric ground load profile (Figure 4).

The comparison is done with the exact solution (no aggregation), the load aggregation algorithm of Yavuzturk (1999), and the two aggregation algorithms proposed by Bernier (Bernier 2001; Bernier et al. 2004). The details of these schemes are given in the references, but we can quickly give the major principles. In Yavuzturk (1999), the hourly loads are aggregated into load blocks except in the short-time history of the time where the temperature is evaluated. In that period, which he called the minimum hourly history period (MHHP), no aggregation is done. Of course, the simulation time and the error induced by the aggregation depend on the two parameters chosen. We kept the parameters proposed by the authors, which are aggregated blocks of 730 hours (one month) and an MHHP of 192 hours. Bernier proposed a simple load aggregation algorithm (SLAA) and a multiple load aggregation algorithm (MLAA). In the first case, only one aggregation block is used where a mean value of all the hourly loads is considered. The size of this block is the difference between the hour considered and the short-term thermal history (STTH). The value of STTH will directly affect the CPU time and the error introduced. Bernier (2001) discusses the effect of this parameter. We kept the one that he suggested (STTH = 4990 hours). Finally the MLAA define five blocks. One block, called the "hourly thermal history" (Xh), is computed without aggregation. The other four blocks are computed with aggregation: a daily aggregated period (Xd), a weekly aggregated period (Xw), a monthly aggregated period (Xm), and, finally, a yearly aggregated period (Xy). The authors discuss the effect of the different numbers chosen for these periods. We only kept the final one proposed by the author: Xh = 12 hours, Xd = 24 hours, Xw = 168 hours, and Xm = 720 hours.

[FIGURE 2 OMITTED]

[FIGURE 3 OMITTED]

The simulations were performed on a pentium 4, 2 GHz, using MATLAB[R] scripts. The CPU-time for the asymmetric profile is listed in Table 1 and shown in Figure 5. The CPU-time for the symmetric case is only listed in Table 2 since the values are similar to the other case. The RMS error is listed in Table 3 for the asymmetric case in and Table 4 for the symmetric case. Finally, the maximum error is listed in Table 5 and shown in Figure 6 for the asymmetric case and in Table 6 and Figure 7 for the symmetric profile.

[FIGURE 4 OMITTED]

[FIGURE 5 OMITTED]

As seen in the results, the aggregation scheme can give very accurate results if they keep a sufficient number of non-aggregation period hours. This can have an appreciable effect on the simulation time. As soon as the aggregation period diminishes, the precision decreases. In the nonhistory-dependent scheme, the real load is always used; the error comes from the numerical evaluation of the improper integral. This aspect can still be improved, but the first results are still very good for the large reduction of simulation time obtained.

[FIGURE 6 OMITTED]

[FIGURE 7 OMITTED]

CONCLUSION

Many models are proposed in the literature for the analysis and design of borehole performance in a GCHP system. Hourly load simulations for simulation time of a year or more are very time consuming, and some aggregation schemes were proposed to reduce this time. When very good accuracy is needed, the simulation time is still important if many simulations are performed. Using a nonhistory-dependent scheme, we achieve an accurate hourly calculation with a much faster scheme than the aggregation schemes. The approach is applied in this paper on a mathematical model using the cylindrical heat source approach with only one borehole, but we will investigate in the future the possibility of using other physical models as well.

REFERENCES

Bernier, M. 2001. Ground-coupled heat pump system simulation. ASHRAE Transactions 106(1):605-16.

Bernier, M., P. Pinel, R. Labib, and R. Paillot. 2004. A multiple load aggregation for annual hourly simulations of GCHP systems, HVAC & R Research 10(4):471-87.

Carslaw, H., and J. Jaeger. 1947. Conduction of Heat in Solids. Oxford: Claremore Press.

Cole, K.D. 2000. Library of Green's functions for heat conduction, www.engr.unl.edu/~glibrary/home/.

Eskilson, P. 1987. Thermal analysis of heat extraction systems. PhD dissertation, Lund University, Sweden.

Greengard, L., and J. Strain. 1990. A fast algorithm for the evaluation of heat potentials. Comm on Pure and Applied Mathematics 43:949-63.

Kavanaugh, S. 1985. Simulation and experimental verification of vertical ground-coupled heat pump systems. PhD dissertation, Oklahoma State University.

Ozisik, M. 1993. Heat Conduction, 2nd ed. New York: John Wiley.

Pinel, A. 2003. Amelioration, validation et implantation d'un algorithme de calcul pour evaluer le transfert thermique dans les puits verticaux de systemes de pompes chaleur geothermiques, master's thesis, Ecole Polytechnique de Montreal, Canada.

Shonder, J., V. Baxter, J. Thornton, and P. Hughes 1999. A new comparison of vertical ground exchanger design methods for residential applications. ASHRAE Transactions 105(2):1179-88.

Yavuzturk, C. 1999. Modeling of vertical ground loop heat exchangers for ground source heat pump systems. PhD dissertation, Oklahoma State University.

Yavuzturk, C., and J. Spitler. 1999. A short time step response factor model for vertical ground loop heat exchangers. ASHRAE Transactions 105(2):475.

Zeng, H., N. Diao, and Z. Fang 2002. A finite line-source model for boreholes in geothermal heat exchangers. Heat Transfer Asian Research 31(7):558-67.

Louis Lamarche, PhD, PE

Benoit Beauchamp

Student Member ASHRAE

Louis Lamarche is a professor and Benoit Beauchamp is a graduate student in the Department of Mechanical Engineering, Ecole de Technologie Superieure, Montreal, Quebec, Canada.
```Table 1. CPU-Time, Asymmetric Profile

No
Simulation Number Aggregation New Bernier Bernier
Time of Hours Scheme Scheme Yavuzturk No. 1 No. 2

1 month 730 11.5 0.23 11.5 11.5 1.1
3 months 2190 109 0.33 49 109 3.3
6 months 4380 444 0.45 105 444 6.1
1 year 8760 1623 0.97 207 1372 11.8
2 years 17,520 6750 1.39 460 3378 25.1

Table 2. CPU-Time (seconds), Symmetric Profile

No
Simulation Number Aggregation New Bernier Bernier
Time of Hours Scheme Scheme Yavuzturk No. 1 No. 2

1 month 730 12.4 0.26 12.4 12.4 1.4
3 months 2190 111 0.34 53 111 3.0
6 months 4380 462 0.49 112 462 6.5
1 year 8760 1640 0.61 216 1382 11.7
2 years 17,520 6372 1.01 430 3287 25.5

Table 3. RMS Error with the Reference Scheme, Asymmetric Profile

Simulation Number New Scheme Yavuzturk
Time of Hours ([degrees]C/[degrees]F) ([degrees]C/[degrees]F)

1 month 730 0.023/0.041 --
3 months 2190 0.023/0.041 0.020/0.036
6 months 4380 0.051/0.092 0.024/0.043
1 year 8760 0.051/0.092 0.024/0.043
2 years 17,520 0.053/0.095 0.023/0.041

Simulation Bernier No. 1 Bernier No. 2
Time ([degrees]C/[degrees]F) ([degrees]C/[degrees]F)

1 month -- 0.08/0.14
3 months -- 0.10/0.18
6 months -- 0.22/0.40
1 year 0.010/0.018 0.24/0.43
2 years 0.033/0.059 0.27/0.49

Table 4. RMS Error with the Reference Scheme, Symmetric Profile

Simulation Number New Scheme Yavuzturk
Time of Hours ([degrees]C/[degrees]F) ([degrees]C/[degrees]F)

1 month 730 0.071/0.128 -
3 months 2190 0.054/0.097 0.017/0.031
6 months 4380 0.054/0.097 0.030/0.054
1 year 8760 0.054/0.097 0.031/0.056
2 years 17,520 0.055/0.099 0.032/0.058

Simulation Bernier No. 1 Bernier No. 2
Time ([degrees]C/[degrees]F) ([degrees]C/[degrees]F)

1 month - 0.24/0.43
3 months - 0.20/0.36
6 months - 0.26/0.47
1 year 0.014/0.025 0.33/0.59
2 years 0.048/0.086 0.37/0.67

Table 5. Maximum Absolute Error with the Reference Scheme, Asymmetric
Profile

Simulation Number New Scheme Yavuzturk
Time of Hours ([degrees]C/[degrees]F) ([degrees]C/[degrees]F)

1 month 730 0.09/0.16 --
3 months 2190 0.11/0.20 0.06/0.11
6 months 4380 0.22/0.40 0.08/0.14
1 year 8760 0.22/0.40 0.08/0.14
2 years 17,520 0.22/0.40 0.08/0.14

Simulation Bernier No. 1 Bernier No. 2
Time ([degrees]C/[degrees]F) ([degrees]C/[degrees]F)

1 month -- 0.25/0.45
3 months -- 0.42/0.76
6 months -- 0.86/1.55
1 year 0.01/0.02 0.87/1.57
2 years 0.07/0.126 0.87/1.57

Table 6. Maximum Absolute Error with the Reference Scheme, Symmetric
Profile

Simulation Number New Scheme Yavuzturk
Time of Hours ([degrees]C/[degrees]F) ([degrees]C/[degrees]F)

1 month 730 0.21/0.38 --
3 months 2190 0.21/0.38 0.07/0.13
6 months 4380 021/0.38 0.09/0.16
1 year 8760 021/0.38 0.09/0.16
2 years 17,520 0.21/0.38 0.09/0.16

Simulation Bernier No. 1 Bernier No. 2
Time ([degrees]C/[degrees]F) ([degrees]C/[degrees]F)

1 month -- 0.62/1.12
3 months -- 0.62/1.12
6 months -- 0.99/1.78
1 year 0.05/0.09 1.01/1.82
2 years 0.10/0.18 1.01/1.82
```
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