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Estimation of soil and grout thermal properties for ground-coupled heat pump systems: Development and application
Applied Thermal Engineering 143 (2018) 112–122
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Applied Thermal Engineering journal homepage: www.elsevier.com/locate/apthermeng
Estimation of soil and grout thermal properties for ground-coupled heat pump systems: Development and application Linfeng Zhanga,b, Jiayu Chena, Junqi Wangc, Gongsheng Huanga,
T
⁎
a
Department of Architecture and Civil Engineering, City University of Hong Kong, Kowloon, Hong Kong College of Civil Engineering, Hunan University, Changsha, Hunan 410082, China c Division of Building Science and Technology, City University of Hong Kong, Kowloon, Hong Kong b
H I GH L IG H T S
proposed method is an effective method for the ground parameters estimation. • The required TRT duration can be greatly shortened by the proposed method. • The estimation accuracy varies significantly for various U-pipe shank spacing. • The relative sizing error for the hourly simulation based design method is 4.49%. • The • Higher grout thermal conductivity will result in a shorten borehole length.
A R T I C LE I N FO
A B S T R A C T
Keywords: Thermal response test Grout thermal conductivity and diffusivity Thermal parameter estimation Ground-coupled heat pump system
Ground thermal properties, including the thermal conductivity and diffusivity of both soil and grout, are significant considerations for the design of a ground-coupled heat pump (GCHP) system. However, as a result of the limitations inherent in available response models, few in-situ thermal response tests (TRTs) can identify grout thermal conductivity and diffusivity. This paper proposes a new method to estimate the thermal conductivity and diffusivity of both soil and grout simultaneously using the recently developed infinite composite-medium line source (ICMLS) model. Firstly, a linear dependence analysis is performed on the aforementioned four parameters to ensure the feasibility of the proposed method, leading to an estimation of the minimum TRT duration. Secondly, uncertainty analysis is carried out to analyze the influence of U-pipe shank spacing, as it is considered a sensitive parameter in modeling the heat transfer of ground heat exchangers (GHEs). Thirdly, a genetic algorithm is used to identify these four parameters using the data collected from a TRT. The proposed method is verified using a well-designed sandbox experiment. Finally, its application is demonstrated and evaluated by applying it to the design of a GCHP system for an office building at Hunan University.
1. Introduction The ground heat exchanger (GHE), a key component in a groundcoupled heat pump (GCHP) system, can be used to fulfill the heating and cooling demand of buildings and promote building energy efficiency. Ground thermal properties such as thermal conductivity and diffusivity of both soil and grout are essential considerations in the design of GCHP systems because they directly affect the system configuration (e.g., the total length) of the GHEs [1]. Along with the design development of GCHP systems, several methods of estimating ground thermal properties have come into use. In the design method based on rules of thumb, only the heat flux of the
⁎
test borehole is needed, and it can be measured directly through a thermal response test (TRT) [2,3]. When the ASHRAE method is used to size the GHE, the thermal conductivity and diffusivity of soil and borehole thermal resistance must be identified. They can be estimated by matching the GHE heat transfer model to the measured fluid temperature datasets from the TRT [4,5]. To adopt more accurate design methods, such as the improved ASHRAE method developed by Li et al. [6,7], the thermal properties of both the soil and grout are necessary. However, it is difficult to quantitatively estimate the grout thermal properties. Until now, only numerical and empirical methods have been available [8]. The numerical method uses a 3D numerical model, which is coded
Corresponding author. E-mail address: [email protected] (G. Huang).
https://doi.org/10.1016/j.applthermaleng.2018.07.089 Received 21 April 2018; Received in revised form 12 June 2018; Accepted 16 July 2018 Available online 18 July 2018 1359-4311/ © 2018 Elsevier Ltd. All rights reserved.
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Nomenclature
rb ri ro R Rb RFLS RICMLS RILS Rp Re t T T0 Texp Tf Tf , ICMLS Tin Tout Tsim V
List of abbreviations COP EER FLS GHEs GCHP ICMLS ILS RSCs TRT
coefficient of performance energy efficiency ratio finite line source ground heat exchangers ground-coupled heat pump infinite composite-medium line source infinite line source relative sensitive coefficients thermal response test
List of symbols
c erf(x ) exp D G ΔG hf H I (x1, x2) ierf(x ) Ji Ji∗ LD m M N12, Ns1 P Pr q Q Qc Qh Qlc Qlh r r+
fluid heat capacity, J/kg °C error function exponential function half U-pipe shank spacing, m G-function response factor thermal interference for the whole borehole field convective heat transfer coefficient the depth of the borehole, m special function defined in Eq. (A.7) integral of the error function erf(x ) sensitive coefficient relative sensitive coefficient dimensionless vertical coordinate of the pipe, m mass flow rate, kg/s numbers of boreholes dimensionless thermal resistance in Eq. (A.12) parameters under consideration Prandtl number heat flux per meter, W/m input power for the TRT, W heat injected into the ground, W heat rejected from the ground, W cooling load of the building, W heating load of the building, W radius, m the radius of the point in the pipe wall, m
the radius of the borehole, m inside radius of the pipe, m outside radius of the pipe, m thermal resistance, m °C/W borehole thermal resistance, m °C/W thermal resistance for FLS model, m °C/W thermal resistance for ICMLS model, m °C/W thermal resistance for ILS model, m °C/W pipe-fluid thermal resistance, m °C/W Reynolds number time, s temperature, °C undisturbed soil temperature, °C measured fluid temperature, °C mean fluid temperature, °C mean fluid temperature calculated by ICMLS model, °C inlet fluid temperature, °C outlet fluid temperature, °C simulated fluid temperature, °C volume fluid rate, L/s
List of greek letters
a αb αp αs κb κf κp κs θ θ+
thermal diffusivity ratio between the ground and grout grout thermal diffusivity, m2/s thermal diffusivity of pipe, m2/s ground thermal diffusivity, m2/s grout thermal conductivity, W/m °C thermal conductivity of fluid, W/m °C thermal conductivity of pipe, W/m °C ground thermal conductivity, W/m °C angle angle at the point in the pipe wall
List of subscripts
b f g p s
borehole fluid grout pipe soil
diffusivity. With the help of linear dependence analysis, we also studied the required TRT duration. Second, an uncertainty analysis of the Upipe shank spacing was done, as it is considered to be a sensitive parameter in GHEs modeling. Finally, a genetic algorithm was used to identify four ground thermal parameters by matching the collected fluid temperature from a TRT to the short-term response model. This proposed method was then verified by a well-designed sandbox experiment, and it was also implemented in a real GCHP system design procedure for an office building at Hunan University. Based on the hourly simulation based design method, as verified by the design tool [18], we analyzed the impact of the estimated ground thermal properties on the GCHP sizing result, especially the grout thermal conductivity. Based on the analysis, the guidelines for future GCHP system design was also provided.
by computational software, such as COMSOL [9], Fluent [10], and TRNSYS [11,12], to calculate the fluid temperature, and soil/grout thermal properties can thus be estimated by matching the measured and calculated fluid temperature. Although the accuracy of the estimated results is usually acceptable, this numerical method is too complex and inflexible to permit modeling of various forms of GHEs [13,14]. Also, the 3D numerical method relies on the accuracy of the borehole’s inside geometry. In contrast, the empirical method assumes that the grout thermal properties are equal to the values of the soil thermal properties or to the values pre-tested in the laboratory [8]. However, this method may lead to inaccuracies in GHE sizing or in performance simulation due to potential large errors between the assumed grout thermal properties and the grout thermal properties on site. Therefore, this paper proposes a new method of estimating ground thermal properties for the in-situ TRT with the short-term response model developed by Li et al. [13,15–17]. The proposed method included three steps. First, to ensure the feasibility of the proposed parameter estimation method, a linear dependence analysis was carried out on the four ground thermal parameters: the soil thermal conductivity and diffusivity and the grout thermal conductivity and
2. Methodology of estimating thermal conductivity and diffusivity of soil and grout 2.1. Overview Given TRT data, the proposed methodology of estimating thermal 113
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the sensitive coefficient should be examined before the estimation program is attempted [19]. In the adopted ICMLS model, the sensitive coefficients are defined as the first partial derivatives of the fluid temperature with respect to the ground thermal parameters [21], which are shown as
conductivity and diffusivity is shown in Fig. 1. To identify the thermal conductivity and diffusivity of both soil and grout, we adopted the infinite composite-medium line source (ICMLS) model developed by Li et al. [13] as the short-term response model. According to inverse heat transfer theory [19], parameter estimation should be carried out when the sensitive coefficients of every parameter under consideration are linearly independent. Therefore, we performed a linear dependence analysis on these four parameters to ensure the feasibility of the proposed parameter estimation method, based on which a minimal TRT duration was estimated. We then carried out uncertainty analysis to analyze the influence of shank spacing, which is considered as a sensitive parameter in modeling of the heat transfer of GHE. Finally, using the data collected from a TRT, the genetic algorithm-based parameter estimation was used to identify these four parameters.
Ji =
Ji = lim
ΔPi → 0
×
Ji∗ = Ji × Pi
+∞
(1) where
2.4. Algorithm for ground thermal properties estimation
⎧ φ = akJ2n (v ) J ′2n (av )−J ′2n (v ) J2n (av ), ψ = akJ2n (v ) Y ′2n (av ) ⎪ −J ′2n (v ) Y2n (av ) ⎪ ⎪ f = akY (v ) J ′ (av )−Y ′ (v ) J (av ), g = akY (v ) Y ′ (av ) 2n 2n 2n 2n 2n 2n ⎪ ⎪ −Y ′2n (v ) Y2n (av ) ⎨ 1 1 ro ⎪ a = ab / as , k = ks / kb, v = urb, Rp = 4πkp ln ri + 4πri hi , hf ⎪ ⎪ = 0.023 kf Re 0.8Pr 0.3 2ri ⎪ ⎪ σ = (kb−ks )/(kb + ks ) ⎩
Fig. 3 is a flowchart of the ground parameter identification algorithm. The identification is a single-objective optimization problem, in which the object function is defined as the agreement between the calculated and measured fluid temperature. The agreement is evaluated using the root mean square error, which is defined by
f=
+∞
+∞
cos[n (θ−θ′)]
0 n =−∞
1 n
n
∑ (Tsim,i−Texp,i )2 i=1
(6)
With the inputs, such as the fluid temperature and the basic information about the borehole, the initial population is generated, which consists of a series of individual variables, including the soil thermal conductivity, the soil thermal diffusivity, the grout thermal conductivity, and the grout thermal diffusivity. The fitness of the individual variables is estimated using Eq. (6). According to the fitness of the individual variables, the best ones are selected from the parent population, and an offspring population is generated by migrating and
Compared with the infinite line source (ILS) model used in the standard estimation method, this model considers the grout thermal capacity inside the borehole, which enables identification of the grout thermal properties based on the inverse heat transfer theory. The ICMLS model used in this paper is slightly different from the original model in Eq. (1), which has the following updated form [20]: 2π
(5)
Note that the estimation of parameter Pi is extremely difficult when is relatively small, as a large range of the Pi values will result in a similar fluid temperature. Furthermore, if one of the RSCs can be written as the linear combination of other RSCs, the parameters cannot be simultaneously restored, as only various combinations of the parameters can be identified. For these two cases, |J ∗T J ∗| ≈ 0 . Thus, a large magnitude of |J ∗T J ∗| is needed for the parameter estimation problem [19].
∑ ∫ [1−exp (−u2αb t )] n =−∞ 0
∫ ∑
(4)
Ji∗
(J2n (urA) + J2n (urB )) J2n (uD)[φg−ψf ] ⎞ du + Rp / N ⎟ u (φ2 + ψ2) ⎠
⎛ 1 Tf (t )−T0 = q⎜ 2 4π kb ⎝
f (P1, ...,Pi + ΔPi, ...,PN )−f (P1, ...,Pi, ...,PN ) ΔPi
To alleviate the difficulties inherent in linear independence analysis of parameters with different orders of magnitude, we used the relative sensitive coefficients (RSCs). The RSCs are defined as
The adopted ICMLS model is defined as Eq. (1) by Li et al. [16]
⎛ 1 Tf , ICMLS (t )−T0 = q⎜ 4πkb ⎝
(3)
Because the heat transfer model T (P ) is complicated, the sensitive coefficients can be simplified using an infinite difference method and become
2.2. Heat transfer model for GHEs
+∞
∂T (P ) , P = {κs, as , κb, ab} ∂Pi
∫ [1−exp(−u2αb t )] 0
J2n (ur +) J2n (uD)[φg−ψf ] ⎞ dudθ+ + Rp / N ⎟ × u (φ2 + ψ2) ⎠
(2)
where + + ⎧ θ = arctan[ro ∗sin(θ )/(D + ro ∗cos(θ ))] ⎨ r + = ro ∗sin(θ+)2 + (D + ro ∗cos(θ+))2 ⎩
Compared with the model described by Eq. (1), which uses the mean Gfunction response factor at points A and B to represent the mean Gfunction response factor at the pipe wall (see Fig. 2), the ICMLS model adopted in this paper uses the integral mean value at the pipe wall instead of the simplified mean value. 2.3. Linear independence analysis of model parameters According to inverse heat transfer theory [19], the parameter estimation should be carried out only when the sensitive coefficients of every parameter under consideration are linearly independent. Thus,
Fig. 1. The proposed methodology of estimating soil and grout thermal conductivity/diffusivity. 114
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offspring variables. The estimation calculation algorithm terminates when the generation reaches the predefined maximum generations (400 generations in this paper). The best individuals, which form the solution for the parameter estimation problem, are outputted at the end of the procedure.
3. Description of the validation TRT The well-organized sandbox experiment platform was operated in Oak Ridge National Laboratory with a controlled environment and predefined ground thermal properties [22]. In this experiment, the high-density polyethylene U-pipe was centered horizontally along a 1.8 m × 1.8 m × 18.3 m wooden box. The sand inside the wooden box served as the soil, and bentonite (20% solids) mixed with water served as the grout. The soil and grout thermal conductivity were measured by the non-steady-state thermal probe with an estimated uncertainty of 5%, and they were served as the benchmark in this study. To validate the accuracy of various parameter estimation methods, various TRTs have been operated in this sandbox experiment platform, including the standard test [22] and interrupted test [23]. In this study, the circulating fluid was heated by an approximately 1100 kW electrical heater, pumped into the U-pipe and cooled by the surrounding sand. Every minute, a data logger automatically collected the inlet and outlet temperatures in addition to the flow rate of the fluid as recorded by two thermocouples and one flow meter. The specifications of the sandbox experiment platform are listed in Table 1.
Fig. 2. Assumed borehole configuration.
recombining the selected individuals. The fitness of the individual variables in the offspring population is also accordingly calculated based on Eq. (6). The new parent population is regenerated by replacing the worst individual variables in the old parent population with the best
Fig. 3. Calculation algorithm for the parameter simultaneously estimation procedure. 115
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than 28 h. Compared with estimation results for the 52 h datasets, the relative errors for the soil and grout thermal conductivity were within ± 3% when the TRT length was longer than 28 h. Thus, the TRT time duration can be shortened using the proposed method, and the required test duration should be t =5rb2 ab .
Table 1 Detailed specifications of sandbox experiment. Parameter
Values
Borehole radius (r b ) Borehole length (H) Inner radius of U-pipe (r i ) Outer radius of U-pipe (ro ) Half U-pipe shank spacing (D ) Thermal conductivity of U-pipe (κp )
0.063 m 18.32 m 0.0137 m 0.0167 m 0.0265 m 0.39 W/m K
Undistributed ground temperature (T0 ) Heat flux (Q ) Volume fluid rate (V )
22.0 °C 1056 W 0.197 L/s
4.3. Uncertainty analysis and estimation results of the proposed method The ASHRAE Handbook [5] states that “the exact computation of borehole thermal resistance is somewhat uncertain due to the uncertainty of the U-pipe location even when the U-pipe spacer is installed”, which indicates that the uncertainty of the U-pipe location will lead to an uncertain fluid temperature prediction. To analyze how the uncertainty of the U-pipe location affects the fluid temperature prediction, sensitivity analysis is necessary. The relative sensitive coefficient of the U-pipe shank spacing was calculated by Eq. (7). The results are illustrated in Fig. 7.
4. Validation for the proposed method 4.1. Linear independence analysis for the ground thermal parameters According to inverse heat transfer theory [19], the RSCs and |J ∗T J ∗| are dependent on the unknown parameters pi . Thus, a priori information on the expected unknown parameters is required. In this paper, the expected values for the TRT are shown in Table 2. The RSCs for the ICMLS model were calculated and are illustrated in Fig. 4. Based on the RSCs values in Fig. 4, the determinant of the J ∗T J ∗ was also calculated and is illustrated in Fig. 5. The magnitude of |J ∗T J ∗| for the ICMLS model is 1e10 at the end of the TRT. Thus, we believe that these four ground thermal parameters in the ICMLS model, including the soil/grout thermal conductivity and the soil/grout thermal diffusivity, are linearly independent, and they can be simultaneously identified. It should be noted that compared with the RSCs of the thermal conductivity of the soil/grout, the RSCs of the thermal diffusivity of the soil/grout are relatively smaller; this mismatch may result in a large error boundary for the grout and soil thermal diffusivity determination. This observation has been verified by Li et al. [24]. Fortunately, according to the small RSCs of thermal diffusivity, a relatively accurate fluid temperature prediction can also be obtained for the thermal diffusivity with a large error. Thus, only the accuracies of the soil and grout thermal conductivity are discussed in this paper.
JD∗ =
∂T (D) D ∂D
(7)
Much as with the fluid temperature prediction, the inverse heat transfer problem is also expected to be sensitive with the choice of the U-pipe shank spacing. Furthermore, in this sandbox experiment, the effective U-pipe shank spacing can be calculated by the steady-state borehole thermal resistance with Hellstrom’s method [27] (0.0344 m), which was significantly different from that provided by the experiment developer (0.0265 m) [22]. Thus, the uncertainty and sensitivity of the U-pipe shank spacing was analyzed as follows to find a suitable U-pipe shank spacing. In this analysis, the U-pipe shank spacing was limited to 0.0167 m (U-pipe walls were in contact with each other) to 0.0463 m (U-pipe legs were spread apart and in contact with the borehole wall). Ten different U-pipe shank spacings were used to find the relationship between the parameter estimation results and the U-pipe shank spacing, which are illustrated in Fig. 8. We can observe that the soil thermal diffusivity given by the proposed method varied ± 4.55% from its median, whereas the grout thermal conductivity varied ± 51.62% from its median. Taking two representative U-pipe shank spacings as examples, including the default value given by the experiment developer (0.0265 m) and the effective value calculated by Hellstrom’s method (0.0344 m) [27], the estimated conductivity/diffusivity of the soil and grout are shown in Table 3. We can observe that the accuracy of the estimation varied significantly for different U-pipe shank spacings. For example, the relative error of the grout thermal conductivity was as high as 21.9% for the default U-pipe shank spacing given by experiment developer. It decreased rapidly to 8.91% when the effective U-pipe shank spacing was used. Thus, the ground thermal properties estimation, especially for the grout thermal conductivity, are affected by the U-pipe shank spacing. Following the trend of the grout thermal conductivity in Fig. 8, we can obtain a linear relationship between the estimated grout thermal conductivity and the U-pipe shank spacing, in which a smaller grout thermal conductivity was estimated with a larger U-pipe shank spacing. For a large borehole filled, the U-pipe shank spacing cannot be accurately determined because of the uncertainty in installation. Following Beier’s study [28], a medium U-pipe shank spacing can be
4.2. Required TRT duration for the proposed method Generally, a longer TRT duration provides a more accurate estimation. However, a TRT cannot be carried out over a long time in practice. Austin [25] indicated that the TRT test should last for no less than 50 h when the ILS model is used, and the TRT duration can be shortened by improving the model’s accuracy during the first few hours. As the ICMLS model is considered as an accurate short-term response model [15], the required TRT length is discussed as follows. According to Ozisik and Orlande’s study [19], the TRT should be long enough when the magnitude of the |J ∗T J ∗| is large enough and not increased significantly. However, this statement is qualitative. As the start time for the standard ground thermal parameter estimation method is t = 5rb2 ab [26] (it is 28 h for the sandbox TRT [15]), this paper takes the time t ⩽5rb2 ab as the potential TRT length. As expected, the magnitude of the |J ∗T J ∗| in Fig. 5 reaches up to 1e10 when the TRT duration is longer than 28 h. Although the magnitude of |J ∗T J ∗| continued to grow for longer test durations, such increments were not as significant as the increment in the experimental time t ⩽5rb2 ab . Thus, t =5rb2 ab should be considered as a suitable test duration for the proposed parameter estimation method. To validate this conjecture, an exhaustion method was also used for the sandbox TRT using the datasets of 4 h, 8 h, 12 h, 16 h, etc. Based on the various TRT duration datasets, various estimation results for the sandbox test were calculated and are illustrated in Fig. 6. Fig. 6 shows that the soil thermal conductivity varied significantly before 28 h, whereas it was much more stable when the test duration was longer
Table 2 Expected values for the unknown parameters. Parameters
Values a
Soil thermal conductivity (ks) Soil thermal diffusivity (as)a Grout thermal conductivity (kb)a Grout thermal diffusivity (ab)a a
116
The values are estimated by Beier et al. [22].
2.82 W/m K 8.81e-7 m2/s 0.73 W/m K 1.92e-7 m2/s
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Fig. 4. Relative sensitive coefficients for the ground thermal parameters. Fig. 7. Relative sensitive coefficients for U-pipe shank spacing.
Fig. 5. The determinant of J ∗T J ∗ under various TRT durations.
Fig. 8. Soil and grout thermal conductivities estimated under different U-pipe shank spacings. Table 3 Estimated results for various U-pipe shank spacings. Parameters
ks (W/m K) as (10−7 m2/s) kb (W/m K) ab (10−7 m2/s) a
Benchmark
D = 0.0265 ma
D = 0.0344 m
Values
Values
RE
Values
RE
2.820 8.810 0.730 1.920
3.244 8.708 0.890 1.707
15.04% 1.16% 21.90% 11.12%
3.071 7.115 0.696 1.498
8.91% 19.24% 4.71% 22.00%
D is Half U-pipe shank spacing.
recommended by the ASHRAE Handbook [5], was regarded as a suitable value for a large-borehole field design. To investigate the estimation accuracy, we used two more methods proposed by the previous studies [22,29] for comparison: the standard estimation method with the ILS model and the two-step parameter estimation (TSPE) method with a numerical model. Table 4 summarizes the results of all aforementioned methods. From Table 4, we can find that the grout thermal parameters, which are crucial for accurate GCHP system design, cannot be estimated by the standard method with the ILS model. Thus, the standard method is not dominant compared with
Fig. 6. Thermal conductivity estimation results under various TRT durations.
recommended as the medium U-pipe shank spacing, which may offer the greatest reliability for accurate parameter estimation. Thus, the medium U-pipe shank spacing, which was a conservative value 117
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Table 4 Estimation results for various methods. Parameters
ks (W/m K) as (10−7 m2/s) kb (W/m K) ab (10−7 m2/s) a b
Standard methoda
TSPE methodb
Benchmark
Proposed method
Values
Values
RE
Values
RE
Values
RE
2.82 8.81 0.73 1.92
3.225 9.502 0.778 1.630
14.35% 7.86% 6.58% 15.10%
2.92 25.84 – –
3.55% 193.30% – –
2.4 7.72 0.76 1.6
14.89% 12.37% 4.11% 16.67%
The estimation result is based on the ILS model. The estimation results by the TSPE method are from Bozzoli et al. [29].
demand of the building. Fig. 10 illustrates the design method based on the hourly simulation, which was essentially an iterative process. Based on the input parameters shown in Table 5, the thermal load of the ground and the fluid temperature can be calculated by the heat pump models and the GHEs heat transfer models, and the length of the borehole can be automatically adjusted to make the fluid temperature meet the design criteria. The detailed flowchart for using the estimated ground thermal properties to size the GHEs depth is illustrated in Fig. 11. Because of the considerable investment in the GHEs, the GHE depth should not only meet the requirement of the heating and cooling demand of the building, but also minimize the initial investment of the GCHP system; the GHE sizing issue was therefore converted into an optimal problem. According to Fig. 11, we optimized the GHE length by the dichotomy method. First, with the input parameters, including the hourly building load and the ground thermal properties, the boundary for the borehole depth and the initial maximum mean fluid temperature can be defined by the designers; the initial borehole depth was set as the middle value for the borehole depth boundary. Second, the hourly mean fluid temperature, the hourly inlet and outlet fluid temperatures, and the hourly coefficient of performance (COP) can be calculated by the hourly simulation method for the GCHP system. The detailed hourly simulation method can be found in the Appendix A and [31]. Third, the maximum mean fluid temperature can be obtained with the hourly simulation method. Comparing with predefined design criteria, the boundary of the borehole depth will be automatically adjusted. If a higher mean fluid temperature is obtained, the lower boundary of the borehole depth is replaced by the previous borehole depth; otherwise, the upper boundary is replaced. The optimization process terminates when the maximum mean fluid temperature meets the design criteria, and the borehole depth can be outputted.
the other two methods. Meanwhile, the TSPE method is based on the numerical model, which is inflexible. Thus, the method proposed here is much more attractive than the TSPE method, based on the qualitative analysis. Furthermore, the relative errors for the estimation results were calculated and compared in Table 4. We found that the relative errors of the soil and grout thermal conductivity calculated by the method proposed here resembles that obtained by the TSPE method but with much higher computational efficiency and simplicity. 5. Application of the parameter estimation method 5.1. Basic information for the building and the ground To demonstrate the advantages of the new proposed parameter estimation method, we also used two other parameter estimation methods, the standard method [5] and the Empirical method [8], to design a GCHP system for a real office building at Hunan University [30]. The building operated in a typical office-working schedule: open from 8:00 am to 6:00 pm and closed from 6:00 pm to 8:00 am the next day; open from Monday to Friday and closed on Saturday and Sunday. Furthermore, an in-situ TRT was also performed at Hunan University in August 2012, and the ILS model was applied for the ground thermal parameters estimation. The details about the building and the TRT are outlined in the authors’ previous work [26]. Both the building load and the inlet and outlet temperature for the in-situ TRT are shown in Fig. 9, and the detailed parameters for the building and in-situ TRT are shown in Table 5. 5.2. Hourly simulation-based GCHP system design method According to the ASHRAE method [5] and the improved ASHRAE method [7], the GHEs were designed to meet the heating and cooling
Table 5 Detailed information of the four selected GCHP systems. Parameter
Values
Borehole configuration Initial borehole depth, m Borehole spacing, m Borehole radius, m U-pipe inner radius, m U-pipe outer radius, m Half U-pipe shank spacing, m Net annual average heat transfer to ground, W Design building heating/cooling load, W Operation period of the GCHP, year Borehole interference temperature penalty, °C Part load factor Outlet fluid temperature, °C Inlet fluid temperature, °C Undisturbed ground temperature, °C Flow rate, L/s
13 rows × 16 columns 120 5 0.075 0.014 0.016 0.0375 −199,175 −2,454,198 1 −3.8a 0.245 32 37 17.5 0.5529
a
The temperature penalty is based on the recommended value in the ASHRAE Handbook [5].
Fig. 9. Building load and TRT data for the office building in Hunan University.
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Fig. 10. Flowchart for the hourly simulation-based design method.
Fig. 11. Detailed hourly simulation-based design method.
According to Beier’s study [28,32], the purpose of the in-situ TRT was to provide input parameters for the design calculation of the GCHP system, and the models and assumptions should be the same for the TRT analysis and the design calculations. Thus, the design calculation with the standard parameter estimation result used the ILS model, whereas the design calculation with the proposed parameter estimation result used the ICMLS model. To cover the calculation period for a whole life cycle, the ILS and ICMLS models were improved by the finite line source (FLS) and full-scale models, respectively. As the FLS model
5.3. Ghes sizing using various parameter estimation results As the numerical estimation method is difficult to perform outside the professional software environment, we used the results estimated by the standard method, the empirical method and the proposed method for our GHEs sizing calculation. It should be noted that the grout thermal properties were equal to the soil thermal properties in the empirical estimation method. The detailed estimation results for these three methods are listed in Table 6. 119
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offered a much longer borehole compared with the design tool result. Furthermore, compared with the proposed estimation results, the grout thermal conductivity in the empirical estimation results was 25.22% higher. As expected, the larger grout thermal conductivity resulted in a much shallower borehole; the borehole depth for the empirical estimation results was 19.63% shorter than the proposed estimation results. Besides that, the borehole depth for the empirical estimation result was 16.03% shorter than the design tool result, which may lead to a failure GCHP system design [7].
Table 6 Estimation results for GHEs sizing at the office building in Hunan University. Parameters
Proposed estimation results
Standard estimation results
Empirical estimation results
Rb (m/K W) ks (W/m K) as (10−6 m2/s) kb (W/m K) ab (10−6 m2/s)
– 2.156 4.41 1.61 1.37
0.111 2.153 1.81 – –
0.111 2.153 1.81 2.153 1.81
6. Conclusion
Table 7 Various design borehole depths for the real GCHP system. Methods
Borehole depth (m)
Relative error
Actual Design tool Standard estimation resultsc Empirical estimation results Proposed estimation results
120a 156b 239 131 163
– – 53.21% 16.03% 4.49%
A new method of identifying ground thermal parameters was proposed in this study, which adopted the ICMLS model and thus could identify grout thermal properties directly using TRT data. Based on the predefined soil and grout thermal properties in the sandbox experiment and the office building at Hunan University, the following findings were observed.
• The proposed method is an effective method to estimate ground
a The actual depth of the GHE was based on the heating load, and the imbalance for the heating and the cooling load will be offset by the cooling tower. b The borehole depth of the design tool was based on the proposed estimation results. c The thermal interactions of multiple GHEs for the standard estimation results were considered by the temperature penalty in Table 5, whereas the thermal interactions of multiple GHEs for the other results were based on Eqs. (A.8)–(A.10).
• •
is not acceptable for the first few hours, the widely used ASHRAE method was used to replace the hourly simulation-based design method for the standard parameter estimation results. To investigate the influence of the grout thermal conductivity on the GHE sizing results, the empirical estimation results also utilized the hourly simulation-based design method for the GHE sizing. With the help of the ASHRAE method and the hourly simulationbased design method, three borehole depths for three estimation results were calculated and are listed in Table 7. To investigate the accuracy of the proposed design method, we took the borehole depth simulated by the design tool [18] as the benchmark in this paper, which is considered an accurate result [33]. With the same input parameters, the relative error between the hourly simulation-based design method and the design tool is as small as 4.49%. This observation can demonstrate that the hourly simulation-based design method is an effective design method. Similar to the conclusion for the design result offered by Cullin et al. [33], the ASHRAE method with the standard estimation results
• •
thermal parameters. Compared with the benchmark, the relative error for the soil thermal conductivity is acceptable. The required TRT duration can be greatly shortened by the proposed method. The new recommended required duration of the TRT for the proposed method was t =5rb2 ab based on the sensitive analysis. The medium U-pipe shank spacing, which was the conservative value in the ASHRAE Handbook, is recommended. With the medium U-pipe shank spacing, the relative error for the soil and grout thermal conductivity was under 15%, an acceptable level for engineering. Compared with the design length by the design tool, the hourly simulation-based design method is effective, and the relative error is limited to 4.49%. The grout thermal conductivity can greatly influence the length of the borehole, and a much higher grout thermal conductivity results in a shorter borehole length; thus, the void content of the grout should be reduced, as it may lead to a lower grout thermal conductivity.
Acknowledgement This work was supported by Strategic Research Grant of City University of Hong Kong (No. 7004630) and Campus Sustainability Fund of City University of Hong Kong (No. 6986047).
Appendix A. Hourly simulation method for a GCHP system The fluid temperature throughout the year can be calculated as the product of the heat flux and the thermal resistance, as shown in Eq. (A.1). N −1
T0−Tf (t ) =
∑ Δqi R (t−ti) = (Δq∗R)(t )
(A.1)
i=0
where Δqi = qi + 1−qi . In Eq. (A.1), the heat flux can be calculated by the building load and the heat pump model. The heat pump model can be simplified as a function of the energy efficiency ratio (EER) and the COP that were offered by the manufacturer [34,35]. The EER and COP formulas in this paper are shown as Eq. (A.2). 2 ⎧ EER = 11.02−0.2170Tout + 0.0009Tout 2 ⎨COP = 3.869 + 0.1232Tout + 0.0005Tout ⎩
(A.2)
Based on the EER/COP, the ground load can be calculated with Eq. (A.3).
⎧ Qc = ⎨Qh = ⎩
1 + EER Qlc EER COP − 1 Qlh COP
(A.3)
By dividing the length of the GHEs, the heat flux for the GHEs can be easily obtained with Eq. (A.4).
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qc = Qc /(M × H ); qh = Qh/(M × H )
(A.4)
In addition to the heat flux, the thermal resistance can be calculated by various GHE heat transfer models, such as the ILS model [24,26,36], the FLS model [37,38], the ICMLS model [13], the equivalent-diameter model [39], and the full-scale model [17]. Among these models, only a few can be used for short-term simulations; the full-scale model was accordingly used in this model. The full-scale model can be written as (A.5)
R = RICMLS + RFLS −RILS + ΔG (t ) In Eq. (A.5), the ICMLS model can be found in Eq. (1), whereas the ILS model and the FLS model can be written as 1 4α t ⎧ RILS = Rb + 4πk ⎛ln r 2s −γ ⎞ ⎪ s⎝ b ⎠ ⎨ ∞ 1 I (Hτ , H1 τ ) 2 ⎪ RFLS = Rb + 4πks ∫1/ 4as t exp(−rb τ 2) Hτ 2 dτ ⎩
(A.6)
where H1 is the distance between the ground surface and the top of the GHEs (H1 = 0 in this paper), and I can be defined as Eq.(A.7) according to Lamarche and Beauchamp [38].
⎧ I (x1, x2) = 2·ierf(x1) + 2·ierf(x1 + 2x2)−ierf(2x1 + 2x2)−ierf(x2) x ierf(x ) = ∫0 erf(u) du = x·erf(x )−1/ π [1−exp(−x 2)] ⎨ ⎩
(A.7)
As with the penalty temperature used to handle the thermal interference of the adjacent boreholes in ASHRAE equation [7], the FLS model was also utilized to calculate the thermal interference of the adjacent boreholes. The thermal interference of adjacent boreholes for the jth borehole can be calculated with Eq.(A.8) [40], and the thermal interference for the whole borehole field can be calculated with Eq. (A.9). M−1
ΔGj (t ) =
∑ i=1
ΔG (t ) =
1 M
1 4πκs
M
∞
∫1/
M−1
∑∑ j=1 i=1
4as t
1 4πκs
exp(−Bi2, j τ 2) ∞
∫1/
4as t
I (Hτ , H1 τ ) dτ Hτ 2
exp(−Bi2, j τ 2)
(A.8)
I (Hτ , H1 τ ) dτ Hτ 2
(A.9)
where Bi, j is the distance between the ith borehole and the jth borehole. As the computational process for Eq. (A.9) was too complex, the thermal interference for the whole borehole field was simplified as the thermal interference of two representative boreholes, the external borehole and the central borehole, shown as M−1
I (Hτ , H1 τ ) ⎡ ∑ 1 ∫∞ exp(−Bi2,1 τ 2) dτ ⎤ Hτ 2 ⎥ 1 ⎢ i = 1 4πκs 1/ 4as t ⎥ ΔG (t ) = ⎢ M − 1 2⎢ ∞ ⎥ 1 2 2 I (Hτ , H1 τ ) + − exp( B τ ) dτ ∑ ∫ i c , 2 ⎢ ⎥ 4πκs 1/ 4as t Hτ ⎣ i=1 ⎦
(A.10)
Finally, using the mean fluid temperature and fluid temperature profile approximations [32,36], the inlet and outlet temperature profile can be calculated using Eq. (A.11).
⎧Tout = ⎨ ⎩
p (2)(Tf − T0) + T0 − (p (3) − p (1) − 2p (3))(qH / cm − T0 ) 1 + p (3) − p (1) − 2p (3)
Tin = Tout + qH / cm
(A.11)
where
⎧ p (2) = a = (N12 + Ns1)2−N122 ⎪ (Ns1 + a2) e a2 ⎪ p (1) = C1 = (Ns1 + a2) e a2 − (Ns1 + a1) e a1 ⎨ p (3) = 1−C1 ⎪ ⎪ Θ(LD ) = T T(LD−) −T T0 , LD = Ll , Q = qH in 0 ⎩
(A.12)
Appendix B. Supplementary material Supplementary data associated with this article can be found, in the online version, at https://doi.org/10.1016/j.applthermaleng.2018.07.089.
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Contents lists available at ScienceDirect
Applied Thermal Engineering journal homepage: www.elsevier.com/locate/apthermeng
Estimation of soil and grout thermal properties for ground-coupled heat pump systems: Development and application Linfeng Zhanga,b, Jiayu Chena, Junqi Wangc, Gongsheng Huanga,
T
⁎
a
Department of Architecture and Civil Engineering, City University of Hong Kong, Kowloon, Hong Kong College of Civil Engineering, Hunan University, Changsha, Hunan 410082, China c Division of Building Science and Technology, City University of Hong Kong, Kowloon, Hong Kong b
H I GH L IG H T S
proposed method is an effective method for the ground parameters estimation. • The required TRT duration can be greatly shortened by the proposed method. • The estimation accuracy varies significantly for various U-pipe shank spacing. • The relative sizing error for the hourly simulation based design method is 4.49%. • The • Higher grout thermal conductivity will result in a shorten borehole length.
A R T I C LE I N FO
A B S T R A C T
Keywords: Thermal response test Grout thermal conductivity and diffusivity Thermal parameter estimation Ground-coupled heat pump system
Ground thermal properties, including the thermal conductivity and diffusivity of both soil and grout, are significant considerations for the design of a ground-coupled heat pump (GCHP) system. However, as a result of the limitations inherent in available response models, few in-situ thermal response tests (TRTs) can identify grout thermal conductivity and diffusivity. This paper proposes a new method to estimate the thermal conductivity and diffusivity of both soil and grout simultaneously using the recently developed infinite composite-medium line source (ICMLS) model. Firstly, a linear dependence analysis is performed on the aforementioned four parameters to ensure the feasibility of the proposed method, leading to an estimation of the minimum TRT duration. Secondly, uncertainty analysis is carried out to analyze the influence of U-pipe shank spacing, as it is considered a sensitive parameter in modeling the heat transfer of ground heat exchangers (GHEs). Thirdly, a genetic algorithm is used to identify these four parameters using the data collected from a TRT. The proposed method is verified using a well-designed sandbox experiment. Finally, its application is demonstrated and evaluated by applying it to the design of a GCHP system for an office building at Hunan University.
1. Introduction The ground heat exchanger (GHE), a key component in a groundcoupled heat pump (GCHP) system, can be used to fulfill the heating and cooling demand of buildings and promote building energy efficiency. Ground thermal properties such as thermal conductivity and diffusivity of both soil and grout are essential considerations in the design of GCHP systems because they directly affect the system configuration (e.g., the total length) of the GHEs [1]. Along with the design development of GCHP systems, several methods of estimating ground thermal properties have come into use. In the design method based on rules of thumb, only the heat flux of the
⁎
test borehole is needed, and it can be measured directly through a thermal response test (TRT) [2,3]. When the ASHRAE method is used to size the GHE, the thermal conductivity and diffusivity of soil and borehole thermal resistance must be identified. They can be estimated by matching the GHE heat transfer model to the measured fluid temperature datasets from the TRT [4,5]. To adopt more accurate design methods, such as the improved ASHRAE method developed by Li et al. [6,7], the thermal properties of both the soil and grout are necessary. However, it is difficult to quantitatively estimate the grout thermal properties. Until now, only numerical and empirical methods have been available [8]. The numerical method uses a 3D numerical model, which is coded
Corresponding author. E-mail address: [email protected] (G. Huang).
https://doi.org/10.1016/j.applthermaleng.2018.07.089 Received 21 April 2018; Received in revised form 12 June 2018; Accepted 16 July 2018 Available online 18 July 2018 1359-4311/ © 2018 Elsevier Ltd. All rights reserved.
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Nomenclature
rb ri ro R Rb RFLS RICMLS RILS Rp Re t T T0 Texp Tf Tf , ICMLS Tin Tout Tsim V
List of abbreviations COP EER FLS GHEs GCHP ICMLS ILS RSCs TRT
coefficient of performance energy efficiency ratio finite line source ground heat exchangers ground-coupled heat pump infinite composite-medium line source infinite line source relative sensitive coefficients thermal response test
List of symbols
c erf(x ) exp D G ΔG hf H I (x1, x2) ierf(x ) Ji Ji∗ LD m M N12, Ns1 P Pr q Q Qc Qh Qlc Qlh r r+
fluid heat capacity, J/kg °C error function exponential function half U-pipe shank spacing, m G-function response factor thermal interference for the whole borehole field convective heat transfer coefficient the depth of the borehole, m special function defined in Eq. (A.7) integral of the error function erf(x ) sensitive coefficient relative sensitive coefficient dimensionless vertical coordinate of the pipe, m mass flow rate, kg/s numbers of boreholes dimensionless thermal resistance in Eq. (A.12) parameters under consideration Prandtl number heat flux per meter, W/m input power for the TRT, W heat injected into the ground, W heat rejected from the ground, W cooling load of the building, W heating load of the building, W radius, m the radius of the point in the pipe wall, m
the radius of the borehole, m inside radius of the pipe, m outside radius of the pipe, m thermal resistance, m °C/W borehole thermal resistance, m °C/W thermal resistance for FLS model, m °C/W thermal resistance for ICMLS model, m °C/W thermal resistance for ILS model, m °C/W pipe-fluid thermal resistance, m °C/W Reynolds number time, s temperature, °C undisturbed soil temperature, °C measured fluid temperature, °C mean fluid temperature, °C mean fluid temperature calculated by ICMLS model, °C inlet fluid temperature, °C outlet fluid temperature, °C simulated fluid temperature, °C volume fluid rate, L/s
List of greek letters
a αb αp αs κb κf κp κs θ θ+
thermal diffusivity ratio between the ground and grout grout thermal diffusivity, m2/s thermal diffusivity of pipe, m2/s ground thermal diffusivity, m2/s grout thermal conductivity, W/m °C thermal conductivity of fluid, W/m °C thermal conductivity of pipe, W/m °C ground thermal conductivity, W/m °C angle angle at the point in the pipe wall
List of subscripts
b f g p s
borehole fluid grout pipe soil
diffusivity. With the help of linear dependence analysis, we also studied the required TRT duration. Second, an uncertainty analysis of the Upipe shank spacing was done, as it is considered to be a sensitive parameter in GHEs modeling. Finally, a genetic algorithm was used to identify four ground thermal parameters by matching the collected fluid temperature from a TRT to the short-term response model. This proposed method was then verified by a well-designed sandbox experiment, and it was also implemented in a real GCHP system design procedure for an office building at Hunan University. Based on the hourly simulation based design method, as verified by the design tool [18], we analyzed the impact of the estimated ground thermal properties on the GCHP sizing result, especially the grout thermal conductivity. Based on the analysis, the guidelines for future GCHP system design was also provided.
by computational software, such as COMSOL [9], Fluent [10], and TRNSYS [11,12], to calculate the fluid temperature, and soil/grout thermal properties can thus be estimated by matching the measured and calculated fluid temperature. Although the accuracy of the estimated results is usually acceptable, this numerical method is too complex and inflexible to permit modeling of various forms of GHEs [13,14]. Also, the 3D numerical method relies on the accuracy of the borehole’s inside geometry. In contrast, the empirical method assumes that the grout thermal properties are equal to the values of the soil thermal properties or to the values pre-tested in the laboratory [8]. However, this method may lead to inaccuracies in GHE sizing or in performance simulation due to potential large errors between the assumed grout thermal properties and the grout thermal properties on site. Therefore, this paper proposes a new method of estimating ground thermal properties for the in-situ TRT with the short-term response model developed by Li et al. [13,15–17]. The proposed method included three steps. First, to ensure the feasibility of the proposed parameter estimation method, a linear dependence analysis was carried out on the four ground thermal parameters: the soil thermal conductivity and diffusivity and the grout thermal conductivity and
2. Methodology of estimating thermal conductivity and diffusivity of soil and grout 2.1. Overview Given TRT data, the proposed methodology of estimating thermal 113
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the sensitive coefficient should be examined before the estimation program is attempted [19]. In the adopted ICMLS model, the sensitive coefficients are defined as the first partial derivatives of the fluid temperature with respect to the ground thermal parameters [21], which are shown as
conductivity and diffusivity is shown in Fig. 1. To identify the thermal conductivity and diffusivity of both soil and grout, we adopted the infinite composite-medium line source (ICMLS) model developed by Li et al. [13] as the short-term response model. According to inverse heat transfer theory [19], parameter estimation should be carried out when the sensitive coefficients of every parameter under consideration are linearly independent. Therefore, we performed a linear dependence analysis on these four parameters to ensure the feasibility of the proposed parameter estimation method, based on which a minimal TRT duration was estimated. We then carried out uncertainty analysis to analyze the influence of shank spacing, which is considered as a sensitive parameter in modeling of the heat transfer of GHE. Finally, using the data collected from a TRT, the genetic algorithm-based parameter estimation was used to identify these four parameters.
Ji =
Ji = lim
ΔPi → 0
×
Ji∗ = Ji × Pi
+∞
(1) where
2.4. Algorithm for ground thermal properties estimation
⎧ φ = akJ2n (v ) J ′2n (av )−J ′2n (v ) J2n (av ), ψ = akJ2n (v ) Y ′2n (av ) ⎪ −J ′2n (v ) Y2n (av ) ⎪ ⎪ f = akY (v ) J ′ (av )−Y ′ (v ) J (av ), g = akY (v ) Y ′ (av ) 2n 2n 2n 2n 2n 2n ⎪ ⎪ −Y ′2n (v ) Y2n (av ) ⎨ 1 1 ro ⎪ a = ab / as , k = ks / kb, v = urb, Rp = 4πkp ln ri + 4πri hi , hf ⎪ ⎪ = 0.023 kf Re 0.8Pr 0.3 2ri ⎪ ⎪ σ = (kb−ks )/(kb + ks ) ⎩
Fig. 3 is a flowchart of the ground parameter identification algorithm. The identification is a single-objective optimization problem, in which the object function is defined as the agreement between the calculated and measured fluid temperature. The agreement is evaluated using the root mean square error, which is defined by
f=
+∞
+∞
cos[n (θ−θ′)]
0 n =−∞
1 n
n
∑ (Tsim,i−Texp,i )2 i=1
(6)
With the inputs, such as the fluid temperature and the basic information about the borehole, the initial population is generated, which consists of a series of individual variables, including the soil thermal conductivity, the soil thermal diffusivity, the grout thermal conductivity, and the grout thermal diffusivity. The fitness of the individual variables is estimated using Eq. (6). According to the fitness of the individual variables, the best ones are selected from the parent population, and an offspring population is generated by migrating and
Compared with the infinite line source (ILS) model used in the standard estimation method, this model considers the grout thermal capacity inside the borehole, which enables identification of the grout thermal properties based on the inverse heat transfer theory. The ICMLS model used in this paper is slightly different from the original model in Eq. (1), which has the following updated form [20]: 2π
(5)
Note that the estimation of parameter Pi is extremely difficult when is relatively small, as a large range of the Pi values will result in a similar fluid temperature. Furthermore, if one of the RSCs can be written as the linear combination of other RSCs, the parameters cannot be simultaneously restored, as only various combinations of the parameters can be identified. For these two cases, |J ∗T J ∗| ≈ 0 . Thus, a large magnitude of |J ∗T J ∗| is needed for the parameter estimation problem [19].
∑ ∫ [1−exp (−u2αb t )] n =−∞ 0
∫ ∑
(4)
Ji∗
(J2n (urA) + J2n (urB )) J2n (uD)[φg−ψf ] ⎞ du + Rp / N ⎟ u (φ2 + ψ2) ⎠
⎛ 1 Tf (t )−T0 = q⎜ 2 4π kb ⎝
f (P1, ...,Pi + ΔPi, ...,PN )−f (P1, ...,Pi, ...,PN ) ΔPi
To alleviate the difficulties inherent in linear independence analysis of parameters with different orders of magnitude, we used the relative sensitive coefficients (RSCs). The RSCs are defined as
The adopted ICMLS model is defined as Eq. (1) by Li et al. [16]
⎛ 1 Tf , ICMLS (t )−T0 = q⎜ 4πkb ⎝
(3)
Because the heat transfer model T (P ) is complicated, the sensitive coefficients can be simplified using an infinite difference method and become
2.2. Heat transfer model for GHEs
+∞
∂T (P ) , P = {κs, as , κb, ab} ∂Pi
∫ [1−exp(−u2αb t )] 0
J2n (ur +) J2n (uD)[φg−ψf ] ⎞ dudθ+ + Rp / N ⎟ × u (φ2 + ψ2) ⎠
(2)
where + + ⎧ θ = arctan[ro ∗sin(θ )/(D + ro ∗cos(θ ))] ⎨ r + = ro ∗sin(θ+)2 + (D + ro ∗cos(θ+))2 ⎩
Compared with the model described by Eq. (1), which uses the mean Gfunction response factor at points A and B to represent the mean Gfunction response factor at the pipe wall (see Fig. 2), the ICMLS model adopted in this paper uses the integral mean value at the pipe wall instead of the simplified mean value. 2.3. Linear independence analysis of model parameters According to inverse heat transfer theory [19], the parameter estimation should be carried out only when the sensitive coefficients of every parameter under consideration are linearly independent. Thus,
Fig. 1. The proposed methodology of estimating soil and grout thermal conductivity/diffusivity. 114
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offspring variables. The estimation calculation algorithm terminates when the generation reaches the predefined maximum generations (400 generations in this paper). The best individuals, which form the solution for the parameter estimation problem, are outputted at the end of the procedure.
3. Description of the validation TRT The well-organized sandbox experiment platform was operated in Oak Ridge National Laboratory with a controlled environment and predefined ground thermal properties [22]. In this experiment, the high-density polyethylene U-pipe was centered horizontally along a 1.8 m × 1.8 m × 18.3 m wooden box. The sand inside the wooden box served as the soil, and bentonite (20% solids) mixed with water served as the grout. The soil and grout thermal conductivity were measured by the non-steady-state thermal probe with an estimated uncertainty of 5%, and they were served as the benchmark in this study. To validate the accuracy of various parameter estimation methods, various TRTs have been operated in this sandbox experiment platform, including the standard test [22] and interrupted test [23]. In this study, the circulating fluid was heated by an approximately 1100 kW electrical heater, pumped into the U-pipe and cooled by the surrounding sand. Every minute, a data logger automatically collected the inlet and outlet temperatures in addition to the flow rate of the fluid as recorded by two thermocouples and one flow meter. The specifications of the sandbox experiment platform are listed in Table 1.
Fig. 2. Assumed borehole configuration.
recombining the selected individuals. The fitness of the individual variables in the offspring population is also accordingly calculated based on Eq. (6). The new parent population is regenerated by replacing the worst individual variables in the old parent population with the best
Fig. 3. Calculation algorithm for the parameter simultaneously estimation procedure. 115
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than 28 h. Compared with estimation results for the 52 h datasets, the relative errors for the soil and grout thermal conductivity were within ± 3% when the TRT length was longer than 28 h. Thus, the TRT time duration can be shortened using the proposed method, and the required test duration should be t =5rb2 ab .
Table 1 Detailed specifications of sandbox experiment. Parameter
Values
Borehole radius (r b ) Borehole length (H) Inner radius of U-pipe (r i ) Outer radius of U-pipe (ro ) Half U-pipe shank spacing (D ) Thermal conductivity of U-pipe (κp )
0.063 m 18.32 m 0.0137 m 0.0167 m 0.0265 m 0.39 W/m K
Undistributed ground temperature (T0 ) Heat flux (Q ) Volume fluid rate (V )
22.0 °C 1056 W 0.197 L/s
4.3. Uncertainty analysis and estimation results of the proposed method The ASHRAE Handbook [5] states that “the exact computation of borehole thermal resistance is somewhat uncertain due to the uncertainty of the U-pipe location even when the U-pipe spacer is installed”, which indicates that the uncertainty of the U-pipe location will lead to an uncertain fluid temperature prediction. To analyze how the uncertainty of the U-pipe location affects the fluid temperature prediction, sensitivity analysis is necessary. The relative sensitive coefficient of the U-pipe shank spacing was calculated by Eq. (7). The results are illustrated in Fig. 7.
4. Validation for the proposed method 4.1. Linear independence analysis for the ground thermal parameters According to inverse heat transfer theory [19], the RSCs and |J ∗T J ∗| are dependent on the unknown parameters pi . Thus, a priori information on the expected unknown parameters is required. In this paper, the expected values for the TRT are shown in Table 2. The RSCs for the ICMLS model were calculated and are illustrated in Fig. 4. Based on the RSCs values in Fig. 4, the determinant of the J ∗T J ∗ was also calculated and is illustrated in Fig. 5. The magnitude of |J ∗T J ∗| for the ICMLS model is 1e10 at the end of the TRT. Thus, we believe that these four ground thermal parameters in the ICMLS model, including the soil/grout thermal conductivity and the soil/grout thermal diffusivity, are linearly independent, and they can be simultaneously identified. It should be noted that compared with the RSCs of the thermal conductivity of the soil/grout, the RSCs of the thermal diffusivity of the soil/grout are relatively smaller; this mismatch may result in a large error boundary for the grout and soil thermal diffusivity determination. This observation has been verified by Li et al. [24]. Fortunately, according to the small RSCs of thermal diffusivity, a relatively accurate fluid temperature prediction can also be obtained for the thermal diffusivity with a large error. Thus, only the accuracies of the soil and grout thermal conductivity are discussed in this paper.
JD∗ =
∂T (D) D ∂D
(7)
Much as with the fluid temperature prediction, the inverse heat transfer problem is also expected to be sensitive with the choice of the U-pipe shank spacing. Furthermore, in this sandbox experiment, the effective U-pipe shank spacing can be calculated by the steady-state borehole thermal resistance with Hellstrom’s method [27] (0.0344 m), which was significantly different from that provided by the experiment developer (0.0265 m) [22]. Thus, the uncertainty and sensitivity of the U-pipe shank spacing was analyzed as follows to find a suitable U-pipe shank spacing. In this analysis, the U-pipe shank spacing was limited to 0.0167 m (U-pipe walls were in contact with each other) to 0.0463 m (U-pipe legs were spread apart and in contact with the borehole wall). Ten different U-pipe shank spacings were used to find the relationship between the parameter estimation results and the U-pipe shank spacing, which are illustrated in Fig. 8. We can observe that the soil thermal diffusivity given by the proposed method varied ± 4.55% from its median, whereas the grout thermal conductivity varied ± 51.62% from its median. Taking two representative U-pipe shank spacings as examples, including the default value given by the experiment developer (0.0265 m) and the effective value calculated by Hellstrom’s method (0.0344 m) [27], the estimated conductivity/diffusivity of the soil and grout are shown in Table 3. We can observe that the accuracy of the estimation varied significantly for different U-pipe shank spacings. For example, the relative error of the grout thermal conductivity was as high as 21.9% for the default U-pipe shank spacing given by experiment developer. It decreased rapidly to 8.91% when the effective U-pipe shank spacing was used. Thus, the ground thermal properties estimation, especially for the grout thermal conductivity, are affected by the U-pipe shank spacing. Following the trend of the grout thermal conductivity in Fig. 8, we can obtain a linear relationship between the estimated grout thermal conductivity and the U-pipe shank spacing, in which a smaller grout thermal conductivity was estimated with a larger U-pipe shank spacing. For a large borehole filled, the U-pipe shank spacing cannot be accurately determined because of the uncertainty in installation. Following Beier’s study [28], a medium U-pipe shank spacing can be
4.2. Required TRT duration for the proposed method Generally, a longer TRT duration provides a more accurate estimation. However, a TRT cannot be carried out over a long time in practice. Austin [25] indicated that the TRT test should last for no less than 50 h when the ILS model is used, and the TRT duration can be shortened by improving the model’s accuracy during the first few hours. As the ICMLS model is considered as an accurate short-term response model [15], the required TRT length is discussed as follows. According to Ozisik and Orlande’s study [19], the TRT should be long enough when the magnitude of the |J ∗T J ∗| is large enough and not increased significantly. However, this statement is qualitative. As the start time for the standard ground thermal parameter estimation method is t = 5rb2 ab [26] (it is 28 h for the sandbox TRT [15]), this paper takes the time t ⩽5rb2 ab as the potential TRT length. As expected, the magnitude of the |J ∗T J ∗| in Fig. 5 reaches up to 1e10 when the TRT duration is longer than 28 h. Although the magnitude of |J ∗T J ∗| continued to grow for longer test durations, such increments were not as significant as the increment in the experimental time t ⩽5rb2 ab . Thus, t =5rb2 ab should be considered as a suitable test duration for the proposed parameter estimation method. To validate this conjecture, an exhaustion method was also used for the sandbox TRT using the datasets of 4 h, 8 h, 12 h, 16 h, etc. Based on the various TRT duration datasets, various estimation results for the sandbox test were calculated and are illustrated in Fig. 6. Fig. 6 shows that the soil thermal conductivity varied significantly before 28 h, whereas it was much more stable when the test duration was longer
Table 2 Expected values for the unknown parameters. Parameters
Values a
Soil thermal conductivity (ks) Soil thermal diffusivity (as)a Grout thermal conductivity (kb)a Grout thermal diffusivity (ab)a a
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The values are estimated by Beier et al. [22].
2.82 W/m K 8.81e-7 m2/s 0.73 W/m K 1.92e-7 m2/s
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Fig. 4. Relative sensitive coefficients for the ground thermal parameters. Fig. 7. Relative sensitive coefficients for U-pipe shank spacing.
Fig. 5. The determinant of J ∗T J ∗ under various TRT durations.
Fig. 8. Soil and grout thermal conductivities estimated under different U-pipe shank spacings. Table 3 Estimated results for various U-pipe shank spacings. Parameters
ks (W/m K) as (10−7 m2/s) kb (W/m K) ab (10−7 m2/s) a
Benchmark
D = 0.0265 ma
D = 0.0344 m
Values
Values
RE
Values
RE
2.820 8.810 0.730 1.920
3.244 8.708 0.890 1.707
15.04% 1.16% 21.90% 11.12%
3.071 7.115 0.696 1.498
8.91% 19.24% 4.71% 22.00%
D is Half U-pipe shank spacing.
recommended by the ASHRAE Handbook [5], was regarded as a suitable value for a large-borehole field design. To investigate the estimation accuracy, we used two more methods proposed by the previous studies [22,29] for comparison: the standard estimation method with the ILS model and the two-step parameter estimation (TSPE) method with a numerical model. Table 4 summarizes the results of all aforementioned methods. From Table 4, we can find that the grout thermal parameters, which are crucial for accurate GCHP system design, cannot be estimated by the standard method with the ILS model. Thus, the standard method is not dominant compared with
Fig. 6. Thermal conductivity estimation results under various TRT durations.
recommended as the medium U-pipe shank spacing, which may offer the greatest reliability for accurate parameter estimation. Thus, the medium U-pipe shank spacing, which was a conservative value 117
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Table 4 Estimation results for various methods. Parameters
ks (W/m K) as (10−7 m2/s) kb (W/m K) ab (10−7 m2/s) a b
Standard methoda
TSPE methodb
Benchmark
Proposed method
Values
Values
RE
Values
RE
Values
RE
2.82 8.81 0.73 1.92
3.225 9.502 0.778 1.630
14.35% 7.86% 6.58% 15.10%
2.92 25.84 – –
3.55% 193.30% – –
2.4 7.72 0.76 1.6
14.89% 12.37% 4.11% 16.67%
The estimation result is based on the ILS model. The estimation results by the TSPE method are from Bozzoli et al. [29].
demand of the building. Fig. 10 illustrates the design method based on the hourly simulation, which was essentially an iterative process. Based on the input parameters shown in Table 5, the thermal load of the ground and the fluid temperature can be calculated by the heat pump models and the GHEs heat transfer models, and the length of the borehole can be automatically adjusted to make the fluid temperature meet the design criteria. The detailed flowchart for using the estimated ground thermal properties to size the GHEs depth is illustrated in Fig. 11. Because of the considerable investment in the GHEs, the GHE depth should not only meet the requirement of the heating and cooling demand of the building, but also minimize the initial investment of the GCHP system; the GHE sizing issue was therefore converted into an optimal problem. According to Fig. 11, we optimized the GHE length by the dichotomy method. First, with the input parameters, including the hourly building load and the ground thermal properties, the boundary for the borehole depth and the initial maximum mean fluid temperature can be defined by the designers; the initial borehole depth was set as the middle value for the borehole depth boundary. Second, the hourly mean fluid temperature, the hourly inlet and outlet fluid temperatures, and the hourly coefficient of performance (COP) can be calculated by the hourly simulation method for the GCHP system. The detailed hourly simulation method can be found in the Appendix A and [31]. Third, the maximum mean fluid temperature can be obtained with the hourly simulation method. Comparing with predefined design criteria, the boundary of the borehole depth will be automatically adjusted. If a higher mean fluid temperature is obtained, the lower boundary of the borehole depth is replaced by the previous borehole depth; otherwise, the upper boundary is replaced. The optimization process terminates when the maximum mean fluid temperature meets the design criteria, and the borehole depth can be outputted.
the other two methods. Meanwhile, the TSPE method is based on the numerical model, which is inflexible. Thus, the method proposed here is much more attractive than the TSPE method, based on the qualitative analysis. Furthermore, the relative errors for the estimation results were calculated and compared in Table 4. We found that the relative errors of the soil and grout thermal conductivity calculated by the method proposed here resembles that obtained by the TSPE method but with much higher computational efficiency and simplicity. 5. Application of the parameter estimation method 5.1. Basic information for the building and the ground To demonstrate the advantages of the new proposed parameter estimation method, we also used two other parameter estimation methods, the standard method [5] and the Empirical method [8], to design a GCHP system for a real office building at Hunan University [30]. The building operated in a typical office-working schedule: open from 8:00 am to 6:00 pm and closed from 6:00 pm to 8:00 am the next day; open from Monday to Friday and closed on Saturday and Sunday. Furthermore, an in-situ TRT was also performed at Hunan University in August 2012, and the ILS model was applied for the ground thermal parameters estimation. The details about the building and the TRT are outlined in the authors’ previous work [26]. Both the building load and the inlet and outlet temperature for the in-situ TRT are shown in Fig. 9, and the detailed parameters for the building and in-situ TRT are shown in Table 5. 5.2. Hourly simulation-based GCHP system design method According to the ASHRAE method [5] and the improved ASHRAE method [7], the GHEs were designed to meet the heating and cooling
Table 5 Detailed information of the four selected GCHP systems. Parameter
Values
Borehole configuration Initial borehole depth, m Borehole spacing, m Borehole radius, m U-pipe inner radius, m U-pipe outer radius, m Half U-pipe shank spacing, m Net annual average heat transfer to ground, W Design building heating/cooling load, W Operation period of the GCHP, year Borehole interference temperature penalty, °C Part load factor Outlet fluid temperature, °C Inlet fluid temperature, °C Undisturbed ground temperature, °C Flow rate, L/s
13 rows × 16 columns 120 5 0.075 0.014 0.016 0.0375 −199,175 −2,454,198 1 −3.8a 0.245 32 37 17.5 0.5529
a
The temperature penalty is based on the recommended value in the ASHRAE Handbook [5].
Fig. 9. Building load and TRT data for the office building in Hunan University.
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Fig. 10. Flowchart for the hourly simulation-based design method.
Fig. 11. Detailed hourly simulation-based design method.
According to Beier’s study [28,32], the purpose of the in-situ TRT was to provide input parameters for the design calculation of the GCHP system, and the models and assumptions should be the same for the TRT analysis and the design calculations. Thus, the design calculation with the standard parameter estimation result used the ILS model, whereas the design calculation with the proposed parameter estimation result used the ICMLS model. To cover the calculation period for a whole life cycle, the ILS and ICMLS models were improved by the finite line source (FLS) and full-scale models, respectively. As the FLS model
5.3. Ghes sizing using various parameter estimation results As the numerical estimation method is difficult to perform outside the professional software environment, we used the results estimated by the standard method, the empirical method and the proposed method for our GHEs sizing calculation. It should be noted that the grout thermal properties were equal to the soil thermal properties in the empirical estimation method. The detailed estimation results for these three methods are listed in Table 6. 119
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offered a much longer borehole compared with the design tool result. Furthermore, compared with the proposed estimation results, the grout thermal conductivity in the empirical estimation results was 25.22% higher. As expected, the larger grout thermal conductivity resulted in a much shallower borehole; the borehole depth for the empirical estimation results was 19.63% shorter than the proposed estimation results. Besides that, the borehole depth for the empirical estimation result was 16.03% shorter than the design tool result, which may lead to a failure GCHP system design [7].
Table 6 Estimation results for GHEs sizing at the office building in Hunan University. Parameters
Proposed estimation results
Standard estimation results
Empirical estimation results
Rb (m/K W) ks (W/m K) as (10−6 m2/s) kb (W/m K) ab (10−6 m2/s)
– 2.156 4.41 1.61 1.37
0.111 2.153 1.81 – –
0.111 2.153 1.81 2.153 1.81
6. Conclusion
Table 7 Various design borehole depths for the real GCHP system. Methods
Borehole depth (m)
Relative error
Actual Design tool Standard estimation resultsc Empirical estimation results Proposed estimation results
120a 156b 239 131 163
– – 53.21% 16.03% 4.49%
A new method of identifying ground thermal parameters was proposed in this study, which adopted the ICMLS model and thus could identify grout thermal properties directly using TRT data. Based on the predefined soil and grout thermal properties in the sandbox experiment and the office building at Hunan University, the following findings were observed.
• The proposed method is an effective method to estimate ground
a The actual depth of the GHE was based on the heating load, and the imbalance for the heating and the cooling load will be offset by the cooling tower. b The borehole depth of the design tool was based on the proposed estimation results. c The thermal interactions of multiple GHEs for the standard estimation results were considered by the temperature penalty in Table 5, whereas the thermal interactions of multiple GHEs for the other results were based on Eqs. (A.8)–(A.10).
• •
is not acceptable for the first few hours, the widely used ASHRAE method was used to replace the hourly simulation-based design method for the standard parameter estimation results. To investigate the influence of the grout thermal conductivity on the GHE sizing results, the empirical estimation results also utilized the hourly simulation-based design method for the GHE sizing. With the help of the ASHRAE method and the hourly simulationbased design method, three borehole depths for three estimation results were calculated and are listed in Table 7. To investigate the accuracy of the proposed design method, we took the borehole depth simulated by the design tool [18] as the benchmark in this paper, which is considered an accurate result [33]. With the same input parameters, the relative error between the hourly simulation-based design method and the design tool is as small as 4.49%. This observation can demonstrate that the hourly simulation-based design method is an effective design method. Similar to the conclusion for the design result offered by Cullin et al. [33], the ASHRAE method with the standard estimation results
• •
thermal parameters. Compared with the benchmark, the relative error for the soil thermal conductivity is acceptable. The required TRT duration can be greatly shortened by the proposed method. The new recommended required duration of the TRT for the proposed method was t =5rb2 ab based on the sensitive analysis. The medium U-pipe shank spacing, which was the conservative value in the ASHRAE Handbook, is recommended. With the medium U-pipe shank spacing, the relative error for the soil and grout thermal conductivity was under 15%, an acceptable level for engineering. Compared with the design length by the design tool, the hourly simulation-based design method is effective, and the relative error is limited to 4.49%. The grout thermal conductivity can greatly influence the length of the borehole, and a much higher grout thermal conductivity results in a shorter borehole length; thus, the void content of the grout should be reduced, as it may lead to a lower grout thermal conductivity.
Acknowledgement This work was supported by Strategic Research Grant of City University of Hong Kong (No. 7004630) and Campus Sustainability Fund of City University of Hong Kong (No. 6986047).
Appendix A. Hourly simulation method for a GCHP system The fluid temperature throughout the year can be calculated as the product of the heat flux and the thermal resistance, as shown in Eq. (A.1). N −1
T0−Tf (t ) =
∑ Δqi R (t−ti) = (Δq∗R)(t )
(A.1)
i=0
where Δqi = qi + 1−qi . In Eq. (A.1), the heat flux can be calculated by the building load and the heat pump model. The heat pump model can be simplified as a function of the energy efficiency ratio (EER) and the COP that were offered by the manufacturer [34,35]. The EER and COP formulas in this paper are shown as Eq. (A.2). 2 ⎧ EER = 11.02−0.2170Tout + 0.0009Tout 2 ⎨COP = 3.869 + 0.1232Tout + 0.0005Tout ⎩
(A.2)
Based on the EER/COP, the ground load can be calculated with Eq. (A.3).
⎧ Qc = ⎨Qh = ⎩
1 + EER Qlc EER COP − 1 Qlh COP
(A.3)
By dividing the length of the GHEs, the heat flux for the GHEs can be easily obtained with Eq. (A.4).
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qc = Qc /(M × H ); qh = Qh/(M × H )
(A.4)
In addition to the heat flux, the thermal resistance can be calculated by various GHE heat transfer models, such as the ILS model [24,26,36], the FLS model [37,38], the ICMLS model [13], the equivalent-diameter model [39], and the full-scale model [17]. Among these models, only a few can be used for short-term simulations; the full-scale model was accordingly used in this model. The full-scale model can be written as (A.5)
R = RICMLS + RFLS −RILS + ΔG (t ) In Eq. (A.5), the ICMLS model can be found in Eq. (1), whereas the ILS model and the FLS model can be written as 1 4α t ⎧ RILS = Rb + 4πk ⎛ln r 2s −γ ⎞ ⎪ s⎝ b ⎠ ⎨ ∞ 1 I (Hτ , H1 τ ) 2 ⎪ RFLS = Rb + 4πks ∫1/ 4as t exp(−rb τ 2) Hτ 2 dτ ⎩
(A.6)
where H1 is the distance between the ground surface and the top of the GHEs (H1 = 0 in this paper), and I can be defined as Eq.(A.7) according to Lamarche and Beauchamp [38].
⎧ I (x1, x2) = 2·ierf(x1) + 2·ierf(x1 + 2x2)−ierf(2x1 + 2x2)−ierf(x2) x ierf(x ) = ∫0 erf(u) du = x·erf(x )−1/ π [1−exp(−x 2)] ⎨ ⎩
(A.7)
As with the penalty temperature used to handle the thermal interference of the adjacent boreholes in ASHRAE equation [7], the FLS model was also utilized to calculate the thermal interference of the adjacent boreholes. The thermal interference of adjacent boreholes for the jth borehole can be calculated with Eq.(A.8) [40], and the thermal interference for the whole borehole field can be calculated with Eq. (A.9). M−1
ΔGj (t ) =
∑ i=1
ΔG (t ) =
1 M
1 4πκs
M
∞
∫1/
M−1
∑∑ j=1 i=1
4as t
1 4πκs
exp(−Bi2, j τ 2) ∞
∫1/
4as t
I (Hτ , H1 τ ) dτ Hτ 2
exp(−Bi2, j τ 2)
(A.8)
I (Hτ , H1 τ ) dτ Hτ 2
(A.9)
where Bi, j is the distance between the ith borehole and the jth borehole. As the computational process for Eq. (A.9) was too complex, the thermal interference for the whole borehole field was simplified as the thermal interference of two representative boreholes, the external borehole and the central borehole, shown as M−1
I (Hτ , H1 τ ) ⎡ ∑ 1 ∫∞ exp(−Bi2,1 τ 2) dτ ⎤ Hτ 2 ⎥ 1 ⎢ i = 1 4πκs 1/ 4as t ⎥ ΔG (t ) = ⎢ M − 1 2⎢ ∞ ⎥ 1 2 2 I (Hτ , H1 τ ) + − exp( B τ ) dτ ∑ ∫ i c , 2 ⎢ ⎥ 4πκs 1/ 4as t Hτ ⎣ i=1 ⎦
(A.10)
Finally, using the mean fluid temperature and fluid temperature profile approximations [32,36], the inlet and outlet temperature profile can be calculated using Eq. (A.11).
⎧Tout = ⎨ ⎩
p (2)(Tf − T0) + T0 − (p (3) − p (1) − 2p (3))(qH / cm − T0 ) 1 + p (3) − p (1) − 2p (3)
Tin = Tout + qH / cm
(A.11)
where
⎧ p (2) = a = (N12 + Ns1)2−N122 ⎪ (Ns1 + a2) e a2 ⎪ p (1) = C1 = (Ns1 + a2) e a2 − (Ns1 + a1) e a1 ⎨ p (3) = 1−C1 ⎪ ⎪ Θ(LD ) = T T(LD−) −T T0 , LD = Ll , Q = qH in 0 ⎩
(A.12)
Appendix B. Supplementary material Supplementary data associated with this article can be found, in the online version, at https://doi.org/10.1016/j.applthermaleng.2018.07.089.
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