DEVELOPMENT AND VALIDATION OF A THERMAL NETWORK MODEL TO PREDICT INDOOR OPERATIVE TEMPERATURES IN DRY ROOFPOND BUILDINGS.
Afzal Hossain Arizona State University The Design School, Box 871605 Tempe, AZ 85287-1605
[email protected]
Alfredo Fernandez-Gonzalez University of Nevada, Las Vegas 4505 Maryland Parkway, Box 454018 Las Vegas, Nevada 89154-4018
[email protected]
ABSTRACT
1. INTRODUCTION
This article presents the development and validation of a thermal network model that can be used to predict the average interior operative temperature of a roofpond building based on outdoor climatic data. The study uses data collected from a roofpond test cell during the cooling season of 2009. Measured data for a 15 day period is used to develop the unsteady state heat-transfer model for the cooling mode operation of the test cell. The thermal network model implements a transfer function method with a time lag (∆t) of one hour in order to calculate the indoor operative temperature.
A roofpond system can be defined as a passive solar strategy in which both heating and cooling occur through the use of natural environmental forces (Marlatt, Murray and Squier, 1984). Roofponds mimic the ways in which nature tempers and controls the global climate and are the only passive solar system that has the ability to both heat and cool without additional system components (Hay &Yellott, 1968). From a thermal standpoint, roofponds are strong performers, providing high solar savings fractions, interior temperature stability, enhanced thermal comfort, and very low operational power requirements (Hoffstatter, 1985). Moreover, due to convective heat transfer within the water bags, heat gains or losses are quickly dispersed throughout the roofpond to create a very homogeneous distribution of heat throughout the floor area covered by the system (Haggard, et al., 1975).
The thermal model presented in this article has a moderately strong correlation (R2 = 80.19%) with the actual measured data. Since the indoor operative temperature is strongly influenced by outdoor air temperature and solar radiation, which are correlated with the hours of the day, residuals of the linear regression between calculated and measured indoor temperatures yields a strong daily pattern. A time series model, on the other hand, yields an improved correlation (R2 = 92.79%). Though significantly improved, the residuals were not fully de-trended and the DurbinWatson statistic is found to be 0.321. To further de-trend the residuals of the time series model an Auto Correlation Factor (ACF) and a Partial Auto Correlation Factor (PACF) test is conducted. The test demonstrates that an Auto Regression (AR) model would be the most appropriate to be able to de-trend the seasonality of its residuals. The new seasonal + AR model yields an R2= 97.69%, with an RMSE of 0.144458 and a Durbin-Watson statistic of 1.74. This model was therefore found to be a good fit in predicting the indoor operative temperature for a roofpond building.
Despite the documented energy savings produced by roofponds, and the development of design guidelines produced to encourage the use of roofponds as a heating and cooling strategy, the vast majority of the buildings in the United States use mechanical systems to heat and cool indoor spaces. However, an increased awareness of the negative effects produced by the exacerbated consumption of energy in buildings has brought about renewed interest in roofponds (Naticchia, Fernandez-Gonzalez, & Carbonari, 2007). At the University of Nevada, Las Vegas, a series of studies have been conducted to validate and further characterize the heating and cooling performance of roofpond systems. The goal of such research is to develop a thermal network model that can be used to predict the interior average temperature of a roofpond building based on climatic data.
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2. DESCRIPTION OF PROBLEM STATEMENT AND SCOPE Like most passive design strategies, roofponds are difficult to model. Too many independent variables, mostly climatic parameters, influence the overall performance of a roofpond. However, such model could be of great help to architects and design professionals for quick evaluation of such system during the early schematic design phase. This study intends to develop and validate a thermal network model that can be used to predict the interior average temperature of a roofpond building based on climatic data. The project consists of two distinct phases: i. Development of unsteady state thermal network model; ii. Validation of the thermal model with measure data.
temperature. A two week period between May and June was specifically used to develop the heat transfer model presented in this article. The predicted operative temperature was then tested against the measured temperature to find the correlation and test for patterns in residuals. Since patterns are observed in residuals, a time series model with Fourier series is used to de-trend the pattern.
2.1. Description of the experimental set-up The test cell has interior dimension of 4’-3”x 6’-10” x8’-0” (see Fig. 1). A corrugated metal deck is used as structural ceiling. On top of this ceiling, there is an EPDM liner and above it a9 in.-deep water mass contained within sealed polyethylene bags (Fig. 2). Standard automated garage doors provide the movable insulation for the roofpond (Fig. 3). The Thermacore® garage doors used in this project have an R-value of 11 h-ft2 °F/Btu (1.94 m2 °C/ W) and are suitable for roofpond applications in the U.S. Southwest.
Fig.1: Basic floor plan of the test cell
2.2. Operation of the test cell During the summer of 2009, the movable insulation panels covered the roofpond starting at 6:00 AM every day. The movable insulation panels of both test cells remained “closed” for 13 hours during the daytime to reduce heat gains from incident solar radiation and the hot outdoor air. The movable insulation panels were retracted every evening at 7:00 PM, when the environment was cooler and could begin to absorb the heat gained by the roofpond throughout the day. 2.3. Scope of the study The test cells are located at the Natural Energies Advanced Technologies (NEAT) Laboratory of the University of Nevada, Las Vegas. The experiment was conducted during the cooling season of 2009. Therefore, it is expected that the model will be working best for the summer months and may not be appropriate for use during the heating months. Fig. 2:Longitudinal section of the test cells 3. METHODOLOGY The first phase of this project used data collected from the roofpond test cell to implement unsteady-state thermal heat transfer principles to predict the average interior operative
The implementation of Fourier series improves the correlation and reduced the trend in residuals. Therefore Auto Correlation Factor (ACF) and Partial Auto Correlation Factor (PACF) tests are used to select among AR, MA, and
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ARIMA models. The seasonal model then is summed with the AR / MA / ARIMA model to further reduce the seasonal trend. The three models are them compared to find the best fit model that represents the actual measured data from the roofpond test cell.
Depending on the operational mode of the movable insulation, there are six different convective heat transfer coefficients for the roof assembly. These are presented in Table1.
Fig. 3: Garage door used as the movable insulation
4. TRANSFER FUNCTION HEAT TRANSFER MODEL A transfer function heat transfer model with a time step of one hour was developed using the average ambient air temperature data (Ta) of the two weeks selected for this study. The effects of conduction, radiation and convection are taken into account using basic heat transfer equations. Surface temperature data reflected the effect of radiation on the various surfaces; therefore Sol-Air temperature (Tsol) was not used in the equations. The thermal network used to calculate the average indoor air temperature (Tin) is presented in Figs. 5 and 6.
Fig.5: Thermal network (panel retracted)
To calculate conduction heat transfer, resistances of all the walls, roof and ground were calculated separately. A convective heat transfer coefficient (hcon) was also calculated to better represent the thermal network and to help determine the heat going in and coming out through the building. The heat absorbed by the roofpond from the test cell was of prime concern. Table 1:Direction of heat transfer through roofassembly Position of Panel Closed Closed Closed Closed Open Open
Air space plane upward downward upward downward Exposed to outdoor Exposed to outdoor
Interior air film upward upward downward downward downward upward
Fig. 6: Thermal network (panel open)
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5. ANALYSIS OF RESULTS
Residual Plot t_measured = 2.55611 + 0.930967*t_calc 4
The measured versus calculated indoor temperature is plotted in Fig.8. A simple linear regression of the fitted model is expressed by the following equation:
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Studentized residual
5.1. Simple Regression
t_measured = 2.55611 + 0.930967*t_calc Coefficients: Parameter Estimate
-2
-4
Standard Error
TStatistic
P-Value
2.56
0.69
3.71
0.0002
Slope
0.93
0.02
38.19
0.0000
The P-value for both intercept and the slope is found to be statistically significant. Analysis of Variance: Source Σ of Squares Mean Square 2724.06 668.64 3392.7
0
100
200 row number
300
400
Fig.7b: Model residuals
Intercept
Model Residual Total (Corr.)
0
P-Value
2724.06 1.87
0.0000
Correlation Coefficient = 0.896 R-squared = 80.29 % Durbin-Watson statistic = 0.137
5.2. Time-series model The hourly outdoor average temperature varies with the time of the day. Therefore a frequency (ω) of 24, and angular frequency of [(2*pi)/24]*T was used to capture the daily variation of the outdoor temperature. To that end,sin((2*pi/24)t);cos((2*pi/24)t)and;sin((2*pi/24)t)*cos(( 2*pi/24)t)were introduced in the model. The measured versus the time-series predicted indoor temperature is plotted in Fig.9. A linear multiple regression yields the following relationship between the variables: t_measured= 1.80566 + 0.961454 * t_calc + 0.182754 * sin(w1t) + 0.908795 * cos(w1t) + 0.195899 * sin(w1t) * cos(w1T)
As expected, there is a statistically significant relationship between t_measured and t_calc at the 95.0% confidence level. The correlation coefficient is equal to 0.896058, indicating a moderately strong relationship between the variables (Fig. 7a). The lower value of the Durbin-Watson statistic tests also indicates the pattern in residuals. The residuals versus row order plot (Fig. 7b) reveals the cyclic pattern that is time (day) dependent, which indicates that inclusion of time variable (T) would yield a better relationship and might result in white noise. Plot of Fitted Model t_measured = 2.55611 + 0.930967*t_calc 38
Coefficients: Parameter CONSTANT t_calc sin(w_1t) cos(w_1t) sin(w_1t)*cos( w_1t)
Estimate Standard TStatistic P-Value Error 1.81 0.96 0.18 0.91 0.20
1.31 0.05 0.10 0.05 0.08
1.38 18.94 1.81 17.17 2.51
0.17 0.00 0.07 0.00 0.01
Analysis of Variance: Source Σ of Squares
Df
Mean F-Ratio Square
P-Value
Model
263.23
4
65.81
0.0000
Residual
20.46
163
0.13
Total (Corr.)
283.69
167
t_measured
35
524.21
32 29 26 23 22
25
28
31 t_calc
Fig.7a: Linear correlation
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R-squared = 92.79 % Standard Error of Est. = 0.354 Mean absolute error = 0.282 Durbin-Watson statistic = 0.321
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Since the P-value in the ANOVA table is less than 0.05, there is a statistically significant relationship between the variables at the 95.0% confidence level (Fig. 10a). As expected, R2 has improved by 13 percent, from 82% to 93%.However, a poor Durbin-Watson statistic represents the presence of cyclic residuals (Fig. 10b).The residuals were then tested with an intention to develop ARIMA model, in any combination of AR, I, and MA. In order to
determine the appropriateness of ARIMA model, an ACF and PACF test was performed, yielding the following results. The ACF and PACF of the residuals from the timeseries model (Fig. 11 a & b) reveals that an AR(2) model will best fit the data. Nonetheless, all three AR(1), AR(2), and ARIMA (1,0,1) model were developed to compare the performance of each of them.
Fig.8: Simple correlation between the measured and the calculated temperature
Fig.9: Simple correlation between the measured and time-series calculated temperature
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Plot of t_measured
5.3. Seasonal + AR (1) Model
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The following relationship was calculated from AR (1) model:
observed
29
27
Z_t = ϕ1 Zt-1 + at = 0.8521* Zt-1 - 0.00406
25
23 23
25
27 predicted
29
31
Fig.10a: Time-series correlation
Coef 0.85 - 0.004 - 0.03
SE Coef 0.04 0.01 0.10
T 20.25 - 0.28
P 0.00 - 0.78
However, the constant is found to be insignificant and therefore excluded from the time-series AR (1) model. The model thus yields the following:
Residual Plot
3 2 Studentized residual
Type AR 1 Constant Mean
AR (1): t_measured= 1.80566 + 0.961454 * t_calc + 0.182754 * sin(w1t) + 0.908795 * cos(w1t) + 0.195899 * sin(w1t) * cos(w1T) + ϕ1 Zt-1 + at = 0.8521* Zt-1
1 0 -1 -2 -3 0
30
60
90 row number
120
150
180
This model yields a higher R2 with a significantly higher Durbin-Watson statistic. The residuals also lacks in cyclic pattern (Fig. 12 a & b).
Fig.10b: Model residuals from time-series correlation
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t_measured
29
27
25
23 24
25
26
27 AR-1
28
29
30
90 row number
120
150
180
Fig.12a: AR (1) correlation 3
Fig.11a: Autocorrelation of the residuals
Studentized residual
2 1 0 -1 -2 -3 0
30
60
Fig.12b: AR (1) model residuals 5.4. Seasonal + AR (2) Model The following relationship was calculated from AR (2) model: Fig.11b: Partial Autocorrelation of the residuals
Z_T = ϕ1 ZT-1 + ϕ2 ZT-2 + aT = 1.1417 * ZT-1 - 0.3562 * ZT-2 0.00989
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Type AR 1 AR 2 Constant Mean
Coef 1.14 - 0.36 - 0.002 - 0.01
SE Coef 0.07 0.07 0.01 0.06
T 15.61 - 4.88 - 0.16
P 0.000 0.000 0.88
The constant is found to be statistically insignificant and therefore is not included in the time-series AR (2) model. The model thus yields as the following: AR (2): t_measured= 1.80566 + 0.961454 * t_calc + 0.182754 * sin(w1t) + 0.908795 * cos(w1t) + 0.195899 * sin(w1t) * cos(w1T) + 1.1417 * ZT-1 - 0.3562 * ZT-2 The AR(2) model yields a R2= 97.69% with an improved Durbin-Watson statistic (1.741). The residuals plot shows mostly white noise and loosely cyclic patterns (Fig. 13a& b).
ARIMA (1,0,1): t_measured= 1.80566 + 0.961454 * t_calc + 0.182754 * sin(w1t) + 0.908795 * cos(w1t) + 0.195899 * sin(w1t) * cos(w1T) + 0.7709 * ZT-1 -0.3084 * ZT-2 The ARIMA (1,0,1) model yields somewhat smaller R2 (96.26 %) with a significantly decreased Durbin-Watson statistic (0.8339). The residuals plot reveals the cyclic patterns (Fig. 14 a & b). 5.6. Comparison of the different models Comparison of the three models is consistent with the findings from the ACF and PACF test. AR(2) model yields the best fit and therefore is used for validation. 31
t_measured
29 31
t_measured
29
27
25 27
23 23
25
24
25
26 AR1_MA1
27
28
29
120
150
180
Fig-14a: ARIMA (1,0,1) correlation 23 23
25
27 AR-2
29
31
Fig-13a: AR (2) correlation
3 2 Studentized residual
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Studentized residual
4 2 0
1 0 -1 -2
-2
-3 -4
0
-6 0
30
60
90 row number
120
150
180
30
60
90 row number
Fig-14b: ARIMA (1,0,1) model residuals
Fig-13b: AR (2) model residuals Table2: Comparison of the three models
5.5. Seasonal + ARIMA (1,0,1) Model The following relationship was calculated from ARIMA (1,0,1) model: Z_T = ϕ1 ZT-1 + ω1 ZT-1 + aT = 0.7709 * ZT-1 - 0.3084 * ZT-1 0.00410 Type AR 1 MA1
Coef 0.77 - 0.31
SE Coef 0.06 0.09
T 13.48 - 3.62
P 0.000 0.000
The constant is found to be insignificant and is not included in the time-series ARIMA (1,0,1) model.The model thus yields as the following:
R-squared (%) Mean absolute error Durbin-Watson statistic
Seasonal + AR (1)
Seasonal + AR (2)
97.70 0.15 1.34
97.69 0.14 1.74
Seasonal + ARIMA (1,0,1) 96.26 0.20 0.83
6. VALIDATION OF THE MODEL The Seasonal + AR (2) model: t_measured = 1.80566 + 0.961454 * t_calc + 0.182754 * sin(w1t) + 0.908795 * cos(w1t) + 0.195899 * sin(w1t)* cos(w1T) + 1.1417 * ZT-1 - 0.3562 * ZT-2
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is used to predict indoor average temperature for 10 days (September 1st to 10th).Fig. 15 shows the measured vs. predicted indoor temperature.
7. SUMMARY AND CONCLUSIONS
A linear regression between predicted and measured daily indoor average temperatures yields the following test statistics: R2 = 97.561 %; mean absolute error = 0.4438, Durbin-Watson statistic = 0.322984 (P=0.0000). Though the R2 is higher and the RMSE is acceptably lower, the Durbin-Watson statistic is considerably lower.
The indoor air temperature of a passive building is highly influenced by the outdoor air temperature; hence, it has a strong correlation with the hours of the day. A linear regression between calculated and measured temperatures yields strong daily pattern in their residuals. A time series model therefore is more appropriate to address the cyclic pattern. This approach helped to improve the correlation between the two variables significantly; more importantly, the model de-trended patterns in residuals. However, the model still showed pattern and therefore, ARIMA model was used. An ACF and PACF test demonstrated that a Seasonal +AR(2) model will improve the residuals (i.e., the patterns will be significantly de-trended). The test statistics of Seasonal + AR(1), Seasonal + AR(2), and Seasonal + ARIMA(1,0,1) also revealed the appropriateness of using a Seasonal + AR(2) model and supported the test findings from ACF and PACF.
8. REFERENCES
Fig. 15: Measured and calculated time-series temperature 3
Studentized residual
2 1 0 -1 -2 -3 0
100
200 row number
300
Fig-16: Validation model residuals The plot of the residuals also reveals a trend (Fig. 16). However, the pattern of the residuals is too large to be considered a daily cycle. Rather they might be due to seasonal variation. A larger data set can be helpful in determining and de-trending the pattern.
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(1) Haggard, K., Hay, H., Saveker, D., Edmiston, J., Feldman, J., Hawes, M., et al. (1975).Research Evaluation of a System of Natural Air Conditioning. California Polytechnic State University. San Luis Obispo: California Polytechnic State University. (2) Hay, H., &Yellott, J. (1968).International Aspects of Air Conditioning with Movable Insulation. Solar Energy, 12, 427-438. (3) Hoffstatter, L. S. (1985). Environmental Heat Transfer for Roof Pond Heating. Trinity University. Trinity University, Texas. (4) Marlatt, W. P., Murray, K. A., & Squier, S. E. (1984). Roof Pond Systems. Energy Technology Engineering Center, U.S. Department of Energy. Canoga Park, CA: Energy Systems Group, Rockwell International. (5) Naticchia, B., Fernandez-Gonzalez, A., & Carbonari, A. (2007). Bayesian Network model for the design of roofpond equipped buildings. Energy and Buildings, 39, 258–272.
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