1、OR-05-5-3 Application of Proper Orthogonal Decomposition to Indoor Airflows Basman Elhadidi, PhD H. Ezzat Khalifa, PhD Member ASHRAE ABSTRACT A fast and efJicient computational model based on the method of proper orthogonal decomposition, POD, is devel- oped to predict indoor airflows. This model ha
2、s been applied successfully to a canonical ofice room, which is mechanically ventilated and air conditioned. The results suggest that the model can be applied quickly and eficiently to predict the indoor velocity and temperature distributions inside the ofice, for conditions other than those used in
3、 forming the base cases of the POD scheme, with a reliability ofR2 0.98. Results also suggest that the POD models can be applied to sensorpluce- ment problems and to real-time indoor airflow control. The indoor flow conditions are obtained in seconds compared to the typical CFD run times of hours to
4、 tens of hours. INTRODUCTION Currently, zonal models (Ren and Stewart 2003) are useful for evaluating interzonal airflows and contaminant dispersion in multizone buildings but cannot account for the spatial and temporal distributions of velocity, temperature, or concentration inside a single zone. T
5、hese spatial and temporal distributions are the domain of computational fluid dynamics, CFD (Awbi 1995), or laborious experimental measurements (Yuan et al. 1999). CFD techniques are computationally expensive and time consuming and need experienced users. Although CFD can be initially used to examin
6、e room air distri- bution, it is not economical to use CFD to perform extensive design optimization for a large number of design choices or in real-time, model-based control of contaminant dispersion, particularly during transient emergency situations because of their high demand for computing resou
7、rces and time. Here we propose a novel approach for evaluating indoor airflows based on proper orthogonal decomposition, POD (Holmes et al. 1996). POD has been successfully applied to develop reduced order models in turbulent flows (Lumley 1981; Arndt et al. 1997), simulate internal combustion engin
8、e in-cylinder flows (Fogleman et al. 2003), design simplified flow control mechanisms (Efe and Ozbay 2003; Ly and Tran 2001; Podvin and Lumley 1998), and as a characterlface recognition tool (Everson and Sirovich 1995). POD represents the flow in terms of the most “energetic” characteristic modes (e
9、igenmodes). In the case of indoor airflows, these eigen- modes may represent different diffuser size, location, and operating conditions, different room geometries, or different contaminant release scenarios. To perform this task, a solution set for different design parameters is first obtained expe
10、rimen- tally or by CFD. From these data, “empirical” eigenmodes are then computed and stored. Design engineers can then use these eigenmodes to evaluate indoor flow conditions within the design space at considerably lower expenditure of time and computing resources. Alternatively, the stored modes c
11、an be used in near-real-time (NRT) model-based predictive control of building airflows. This paper is the first in a series that will focus on POD development for indoor airflows. The ultimate goal is to apply POD models to control airflows in large spaces and to predict NRT temperature and contamin
12、ant distributions during tran- sient events. Here we will present the mathematical formula- tion of the POD technique and demonstrate how it can be applied to indoor flows. Then we will demonstrate how POD can be used as (i) a ventilation system design tool and (2) a control sensor (e.g., thermostat
13、) placement tool. In the case of the design tool, we will apply the POD to an office space with a supply vent at an arbitrary location and variable discharge Basman Elhadidi is assistant professor of aeronautical engineering at Cairo University, Egypt. H. Ezzat Khalifa is NYSTAR Distinguished Profes
14、sor of mechanical and aerospace engineering, and director, STAR Center for Environmental Quality Systems, Syracuse University, Syra- cuse, NY. 02005 ASHRAE. 625 velocity. For the control application we will apply the POD N N 1 1 T JN N R, = (ui,uj)=- (ui,u.) = - 1 juiujdx, (3) k= 1 k= 1 method to a
15、nonisothermal ventilation jet that is used to cool the office space. Conclusions and future recommendations will then foilow. PROPER ORTHOGONAL DECOMPOSITION THEORY To construct the POD modes, we consider an ensemble of flow “snapshots,” Ui (x), i = 1,2, . , ., N, where Ui represents a solution set
16、(velocity, temperature, concentration, etc.), x represents the spatial coordinates, and Nis the number of snap- shots. In this work, Ui(x) represents the velocity magnitude (speed), Vi(x), and the temperature, T,(x); hence, Ui(x)=Vi(x),Ti(x) and is a matrix of dimensions Npx2 (where Np is the number
17、 of spatial points). Each snapshot represents the flow field inside the indoor space with different operating conditions and/or different geometry. These snap- shots can be obtained either by experiment or by numerical simulations. Our goal is to represent any one of those snap- shots as N, - Ui(X)
18、= (x) + u;(x) U(X) + c CkPk(X), (1) k= 1 where (x) = (Ui(x) is the ensemble average of the spatial field, and ui(x) is the deviation of a given spatial field from the ensemble average, which can be expanded in terms of the eigenmodes (modes), k(), with ck representing the ampli- tude (relative weigt
19、ing) of each mode. Note that modes k(X) represent both the speed and temperature; hence, modes, (bk(x), such that N, is as small as possible, i.e., we need the minimum number of modes to reconstruct the indoor airflow field to a determined accuracy. These modes are the solution of the classical Fred
20、holm eigenvalue problem (Holmes et al. 1996). This eigenvalue problem can be set up using two approaches (Sirovich 1987): (a) the direct method or (b) the method of snapshots. In the direct method we construct a two-point correlation matrix for every flow vari- able in the ensemble. For three-dimens
21、ional problems this matrix would be of size where N, is the number of flow variables in the ensemble. Typically this is a large number, and determination of the eigenmodes is computation- ally costly. Alternatively, in the method of snapshots, the correlation matrix size is equal to the square of th
22、e number of snapshots, N2 ZI 1 1 0.5 0.5 o O Figure 6 Comparison of the speed (mh) between the reconstructed and original data in aplane cutting through a downward jet. the POD model to the ensemble of data that accounts for the variable vent location on the back wail and then apply the POD model fo
23、r the remainder of the data (variable vent location on the side wall). At reconstruction, we then use the appropriate eigenvalues and eigenmodes depending on the dominant feature. The drawback is that we need to compute the modes twice; the advantage is that the speed and accuracy of the solu- tions
24、 are significantly improved. Figure 5 shows the energy content approaching 100% using 30 modes to reconstruct each solution set. In fact, to maintain the same accuracy as the cases presented above, we only need to retain 10 modes for the case with the inlet diffuser on the back wall and 15 modes for
25、 the other configurations. Since it is possible to segregate the design problems into several smaller problems, we considered including ceiling ventilation as a design alternative. Here we demonstrate the POD technique for the same simplified canonical office, cios- ing the door, and adding a return
26、 vent (O. 15 x O. 15 m) at the center of the ceiling. The supply vent (O. 15 x O. 15 m) is moved in the first quadrant in increments of 0.25 m (0.325 0.99). The main advantage of POD is the reconstruction and assessment of different flow fields other than those used in constructing the base function
27、. Figure 9 shows the values of ck for the first four POD modes. Here ck only depends on two independent variables, the flow rate and temperature. We observe from these figures that the first mode is associated with the largest amplitude and the smoothest coefficient distri- bution (this is similar t
28、o regular Fourier decomposition, where 630 ASHRAE Transactions: Symposia Speed (mts) Speed (mts) x (m) x (m) Temperature (KeMn) Temperaiore (KeMn) ReconSiNcied O 1 2 3 4 Figure 8 Comparison of the speed (mh) and temperature (K) between the reconstructed and original data in a plane along the jet for
29、 the buoyant jet case. 40 10 5 u- 0 UN 0 -5 40 -10 1 1 8 8 AT (K) Speed( however, for the interpolated set, the error decreases to a minimum value, then increases again. The increase is due to the error in interpolating the coefficients ck from the wavy distribution shown in Figure 9. This suggests
30、that we should only include the first three modes of the expansion to reconstruct a solution not existing in the original data set. We can improve this by having a larger number of snapshots in the ensemble such that the coefficients ck are smoother. Figure 11 compares the velocity and temperature d
31、istribution in a cross-sectional plane across the door opening and is deliberately not shown to go through the jet, since the velocities are much lower on that plane (with an expected higher velocity discrepancy). The results in this section can be easily extended to modeling of contaminant concentr
32、ation (a scalar quantity). 632 The transport equations for conccntration resemble those for temperature and are, in fact, easier to model because they are not coupled with the flow field through buoyancy. We will address the POD application for contaminant modeling in a future paper; however, we can
33、 address some issues such as sensor (thermostat) placement using the results we have obtained so far. Let us assume that we wish to determine and control the temperature (or contaminant concentration) at a given point in an indoor space. Two questions arise: (1) where do we place the appropriate sen
34、sors, and (2) how many sensors will we need? Determining the location of the sensor can be addressed by finding points on the walls that have the smallest advection time from a suggested heat (contaminant) source. We will need to construct a cost function to determine the points on the walls that ar
35、e best in minimizing the advection time from different source locations (Arvelo et al. 2002). The second question can be addressed by examining the number of POD modes. For instance, assume that we wish to fix the temperature at agiven value and that we have one sensor in the wall close to the door.
36、 Figure 12 shows the value of a fixed wall temperature for different inlet velocities and temperatures. It is evident from Figure 12 that we can get the same temperature at that location using different supply velocity and tempera- ture conditions as shown by the solid contour. Hence, we cannot pred
37、ict the temperature (or contaminant concentration) and velocity field using this one sensor. However, we can determine with 96% reliability, the indoor airflow and temper- ature (or concentration) field using three sensors on the walls. In this case we express the temperature in terms of three basic
38、 functions and apply a least squares fit to compute the constants ck, k = 1 .3. This same procedure can be applied to contam- inant dispersion in large spaces. A number of sensors based on the number of POD modes can be distributed in a space such that the minimum advection time cost function is sat
39、isfied. This will be addressed in detail in a future paper. ASHRAE Transactions: Symposia Speed (do) ReconsiNded 3 O 25 02 25 2 15 h O 15 h 1 o1 O 05 05 O Temperature or 0.5 296.5 O Speed (mls) Data O 25 0.2 0.15 3 25 2 .-. - E 15 r 1 o1 05 O 0.05 Temperature o() Daia Figure 11 Comparison of the spe
40、ed (mh) and temperature (K) between the reconstructedsolutions not included in the original ensemble and the original CFD data. 1 295- 0.9 0.5 Inflow Temperature (K) Inflow Speed (ms) Figure 12 Locus of ajxed temperature at a point on the wall for diferent infow conditions. ASHRAE Transactions: Symp
41、osia 633 CONCLUSIONS AND RECOMMENDATIONS A POD model for predicting indoor airflows in an office space has been successfully implemented for a number of simple but realistic test cases. Starting with a limited set of experimentally or CFD obtained “snapshots” of the tempera- ture and velocity fields
42、 under different design and operating conditions, the POD model can accurately predict the velocity and temperature fields inside the ventilated space at different design and operating conditions, with reliability of R2 0.99. The model can successfully account for design alternatives not included in
43、 the original survey and, hence, can be used as a design tool to assess the impact of changing diffuser location or varying supply air volume (e.g., in a VAV system) on room air distribution. POD data reconstruction takes seconds on a personal computer, compared to hours for a full CFD simula- tion,
44、 making this an attractive approach for model-based envi- ronmental control applications. The results demonstrate that the POD model can be used to tackle problems of sensor placement for either thermal control or contaminant modeling. The number of dominant modes is a good indication of the number
45、of sensors needed. Furthermore, the POD modes reconstruct the spatial flow features, which cannot be included in currently used control systems. Future work will focus on sensor location problem and on unsteady, near-real-time control situations. The results of this paper suggest that POD models cou
46、ld find fertile ground in thermal environmental control and contaminant dispersion modeling for large spaces, where zonal methods fail to predict the spatial distributions. The model will give the spatial gradients and can be easily adapted to account for the time evolution, as we plan to show in fu
47、ture papers. ACKNOWLEDGMENT The work described in this paper was performed with financial support from the New York State Office of Science, Technology and Academic Research, NYSTAR. REFERENCES Arndt, R.E.A., D.F. Long, and M.N. Glauser. 1997. The proper orthogonal decomposition of pressure fluctua-
48、 tions surrounding a turbulent jet. Journal of Fluid Mechanics 340: 1-33. Arvelo, J., A. Brandt, R.P. Roger, and A. Saksena. 2002. An enhanced multizone model and its application to opti- mum placement of CBW sensors. ASHRAE Transac- tions 108(2). ASHRAE. 2001. 2001 ASHRAE Fundamentals Hand- bookCD.
49、 Atlanta: American Society of Heating, Refrig- erating and Air-conditioning Engineers, Inc. Awbi, H.B. 1995. Ventilation ofBuildings. E & FN Spon, an imprint of Chapman and Hall. Bui-Thanh, T., M. Damodaran, and K. Willcox. 2003. Proper orthogonal decomposition for parametric applications in transonic aerodynamics. AIAA Paper 2003-42 13, 15th Computational Fluid Dynamics Conference, Orlando, FL, June 2003. Chatterjee, S., and B. Price. 1977. Regression Analysis by Example. John Wiley and Sons. Efe, M.O., and H. Ozbay. 2003. Proper orthogonal decompo- sition for reduced order mod
