Simulation and Experimental Validation of Chaotic Behavior of Airflow in a Ventilated Room

Transcription

1 Simulation and Eperimental Validation of Chaotic Behavior of Airflow in a Ventilated Room Jos van Schijndel, Assistant Professor Eindhoven Universit of Technolog, Netherlands KEYWORDS: Airflow, chaos, sstem, ventilation SUMMARY: Chaos ma lead to instabilit, etreme sensitivit and performance reduction in dnamic sstems. Therefore it is unwanted in man cases. Due to these undesirable characteristics of chaos in practical sstems, it is important to recognize such a chaotic behavior. In this paper the chaotic behavior of the airflow in case of an ordinar ventilated room is researched. Computational chaotic behavior is alread observed in the simulations b changing the suppl air temperature from 22 o C into 21.9 o C However, it could be the case that, despite all efforts, the chaotic behavior is a numeral artifact. Therefore laborator eperiments using a scale model were performed. It is concluded that the computational model of the eperimental scale model has to be improved to simulate the eperimental results more accuratel.furthermore, the presented method seems promising in detecting chaotic behaviour. 1. Introduction Chaos theor is a field of stud in mathematics, phsics, economics, and philosoph studing the behavior of dnamic sstems that are highl sensitive to initial conditions. This sensitivit is popularl referred to as the butterfl effect. Small differences in initial conditions (such as those due to rounding errors in numerical computation) ield widel diverging outcomes for chaotic sstems, rendering long-term prediction impossible in general. This happens even though these sstems are deterministic, meaning that their future behavior is full determined b their initial conditions, with no random elements involved. In other words, the deterministic nature of these sstems does not make them predictable. This behavior is known as deterministic chaos, or simpl chaos. Chaotic behavior can be observed in man natural sstems, such as the weather. Eplanation of such behavior ma be sought through analsis of a chaotic mathematical model. Two icons of the chaos theor are shown in Figure 1 (Glendinning (1994); Lorenz, (1963)) FIG 1. Two icons of chaos. Left: Bifurcation diagram of the logistic map. Right: The Lorentz attractor 517

2 Figure 1 left, shows the bifurcation diagram of the logistic map n+1 r n (1 n ). This map represents a discrete model for a dnamic sstem. Each vertical slice shows the attractor (solution of ) for a specific value of r. The diagram displas period-doubling as r increases, eventuall producing chaos in the dnamic sstem. Figure 1 right, presents the Lorenz attractor that also displas chaotic behavior in a dnamic sstem. These two plots demonstrate sensitive dependence on initial conditions within the region of the phase space occupied b the attractors. Moreover, chaotic sstems ma lead to instabilit, etreme sensitivit and performance reduction. Therefore it is unwanted in man cases. Due to these undesirable characteristics of chaos in practical sstems, it is important to recognize such a chaotic behavior. The eistence of chaos has been discovered in several areas during the last 30 ears. However, there is a lack of studies in relation with buildings that also can be regarded as comple dnamic sstems as well. Furthermore, chaotic behavior ma have an enormous impact on the predictabilit of the indoor climate. For eample, if the indoor climate has, under certain circumstances, a similar dnamic behavior as the Lorenz attractor then it could spontaneousl and without an detectable cause turn from hot into cold. In an etreme situation this ma lead to full unstable and unpredictable airflows. The main question is, whether chaotic behavior is rare or more common inside buildings. In this paper research on the chaotic behavior of airflow is presented in case of an ordinar ventilated room with an on/off controller. The approach to detect chaos in a given sstem is to investigate both numerical as well as eperimental results. There are several universal indicators for chaos. In this research we use the propert that all chaotic sstems are etreme sensitive for initial conditions and/or sstem parameters, i.e. small differences in them can lead to etraordinar differences in the sstem states. Our research methodolog was as follows: (1) Literature review on the application of chaos in the built environment. Cai et al. (2006), Elnashaie et al. (2007), Fradkov et al. (2005), Karatasou et al. (2009), Tavazoei et al. (2009) provide a good introduction to the subject. (2) Selection of a specific promising case stud and reproduce simulation results; (3) Investigate computational chaotic behavior using this case stud b small changes in sstem parameters. If promising results are obtained, proceed with: (4) Eperiments in a scale model. The paper is organized in the same wa. 2. Numerical case stud: Airflow in a ventilated room A case stud was selected based on the work of Sinha et al. (2000). The subject of this case stud comprehends airflow in a ventilated room. Figure 2 shows the geometr. FIG 2. The geometr of the airflow problem. The PDEs are based on the well-known Navier-Stokes equations : 518

3 t ( uu) ( vu) p u 1 2 v t ( uv) ( ut) ( vv) ( vt) p Re 1 Re T 1 2 t Re Pr The boundar conditions are: 2 T u v Gr At the left, right, top and bottom walls: u=0, v=0, T=0. At the inlet: At the outlet : u=1, v=0, T=1. Re 2 T u v Neuman conditions for u,v and T The temperature solutions for several Re and Gr numbers from Sinha et al. (2000) are shown:. 0 FIG 3. Temperature inside the room for several Re and Gr according Sinha et al. (2000) (left) and Comsol (right) The most interesting behavior of the airflow in the ventilated room is observed when buoanc is taken into account (i.e. Gr = and Re = 1000). 519

4 Figure 4 shows the temperature distributions after respectivel 20 and 40 time steps in case of the constant inlet air temperature boundar condition. FIG 4. Comple airflow patterns after 20 (left) and 40 (right) time steps. 3. Chaotic behavior using the numerical case stud In order to detect chaotic behavior, the airflow was again simulated with the onl difference that the suppl air temperature was slightl changed from 22 o C into 21.9 o C (i.e. 1 into for the scaled temperature). In Figure 5, the results are presented. FIG 5. Scaled temperature distribution after 60 time steps (case: Gr = and Re = 1000); Top Left: Air suppl temperature equals 1 (22 o C); Top Right: Air suppl temperature equals (21.9 o C); Bottom: Air temperature difference between the top figures where 0 means no difference; +1 means 4 o C hotter; -1 means 4 o C colder 520

5 Figure 5 shows the temperature distribution after 60 time steps. The top left figure has an air suppl temperature of 22 o C and the an air suppl temperature of the top right figure, 21.9 o C The bottom (left) figure represents the distribution in air temperature difference between the two top figures. The results show that temperature distribution seems to be ver sensitive for the air suppl temperature: A small change of 0.1 o C in the air suppl temperature (i.e. 1 into for the scaled temperature) ma temporar and locall lead to opposite temperatures i.e. hot instead of cold and visa versa. This can be best observed in the bottom part of Figure 5. It is clear that chaotic behavior is observed, even without an (on/off) controller which is common in ventilated rooms. We proceed with adding an on/off controller to the previous model. Figure 6 shows the SimuLink model including the on/off controller (Rela) and a so-called S-Function with the Comsol model. Modeling details can be found in van Schijndel (2005, 2007). FIG 6. The SimuLink model with the Comsol model implemented into an S-Function The sensor of the Rela provides input for this SimuLink block and is located at position =2; =2.5 in the room represented b Figure 2. The Rela switches the suppl temperature (scaled) between 1 (hot, 22 o C) and 0 (cold 18 o C) if the sensor temperature is respectivel below 0.3 (i.e o C ) and above 0.5 ( i.e. 20 o C). In order to investigate the occurrence of chaotic behavior analog to the beginning of this Section, a slight change of a sstem parameter is made and differences in airflow patterns are studied. In this case the effect of changing the sensor temperature threshold for switching cold air from 0.3 into 0.32 and hot air from 0.5 into 0.48, is studied. The results obtained from this computational eperiment are comparable with Figure 5. As epected, this produces even greater differences between the two simulations, with just one slightl different parameter setting. The numerical eperiments presented in this Section provide evidence for chaotic behavior, such as small differences in parameters ield widel diverging outcomes. On the other hand, it could be the case that, despite all efforts, the chaotic behavior is a numeral artifact. Therefore we proceed with eperiments with a scale model. 521

6 4. Eperimental case stud: A scale model 4.1 Measurements Figure 7 shows the eperimental scale model, with eternal dimensions 0.64m 0.45m 0.39m (van Goch et al. 2008). Each length scale (i.e.,, z) is a factor 7.7 smaller than the well documented full scale room eperiment of Lu et al. (1997) The dimensions are shown in Figure 7. FIG 7. The eperimental scale model. The sides and top of the bo are constructed as follows: 3 mm Pleiglas 4 mm air cavit 3 mm Pleiglas. The material properties of Pleiglas are: heat conductivit are: 0.21 W/mK; densit, 1190 kg/m3; heat capacit, 1500 J/kgK. The window and bottom are made of 4 and 6 mm Pleiglas respectivel. Three different tpes of measurements were completed: thermal imaging, air flow visualization using smoke, acquisition of temperature sensors over time. Thermal images were obtained of the eternal side including the windows (front). These images are ver useful for validation purposes because the show a 2D distribution over time. Figure 8 (left) shows an image using a heat source of 30 W. FIG 8. Thermal image (left) and visualization of the airflow (right) The previous mentioned thermal images provide some indication of the air flow patterns inside the bo. In order to get more detailed information on the inside air flow, smoke was injected using a small entrance near the bottom of the heating source. Figure 8 (right) shows a tpical result obtained during stead state (after 120 min and 30W input power) conditions. The first step was to reproduce these eperimental results numericall (see net section). The second step would be to stud whether small changes in the eperiment would lead to large effects. 522

7 4.2 Computational model The modeling details are published in van Goch et al. (2008). A short summar is presented below. The combination of the Convection and Conduction and Navier-Stokes application modes of Comsol are used to model and simulate the eperiments. Furthermore onl default grids (course, fine, etc.) and solvers are used. The calculations were limited to a maimum of 32 GB memor at a Sun computer. With the 3D results it is possible to compare thermal images with simulations. The simulated surface temperatures are provided in Figure 9 (right). Figure 10 (left) shows the temperature and velocit after 900 seconds. Figure 10 (right) provides the air circulation FIG 9. Left: Right: Simulated surface temperature FIG 10. Left: The temperature and velocit after 900 seconds; Right: The air circulation After evaluation of all results it has been determined the primar reason for the discrepanc between the results obtained from the simulation and those of the eperiments on the model enclosure was most likel due to the modeling of the heat source. The simulated air velocit is simpl too low just above the heating coil. To achieve the desired agreement between results of the eperimental work and that of the simulation, it is likel better to model the heat source in greater detail (to improve the simulated velocit just above the heat source) or use a velocit profile instead of natural convection. Evidentl, the computational model needs improvement if it is to accuratel simulate the eperimental results; this is the focus of current and on-going research. After this validation our aim is to stud the effect of small changes on both the eperimental scale model as well as the computational model. 523

8 5. Conclusions Chaos ma lead to instabilit, etreme sensitivit and performance reduction in dnamic sstems. Due to these undesirable characteristics of chaos in practical sstems, it is important to recognize such a chaotic behavior. The eistence of chaos has been discovered in several areas during the last 30 ears. However, there is a lack of studies in relation with buildings that also can be regarded as comple dnamic sstems. In this paper the chaotic behavior of the airflow in case of an ordinar ventilated room is researched. Computational chaotic behavior is alread observed in the simulations b changing the suppl air temperature from 22 o C into 21.9 o C However, it could be the case that, despite all efforts, the chaotic behavior is a numeral artifact. Therefore laborator eperiments using a scale model were performed. The first step was to reproduce these eperimental results numericall. We conclude that the computational model has to be improved to simulate the eperimental results more accuratel. The second step will be to stud whether small changes in the eperiment will lead to large effects on both the eperimental scale model as well as the computational model. This could provide direct evidence for the chaotic behavior of the airflow in a small scale room and could be a strong indication that chaotic behavior is perhaps more common inside buildings as epected so far. References Cai, W., Sen, M., Yang, K.T. & McClain, R.L., 2006, Snchronization of self-sustained thermostatic oscillations in a thermal-hdraulic network, Int. Journal of Heat and Mass Transfer 49, pp Elnashaie, S.S.E.H. & Grace, J.R., 2007, Compleit, biconfiguration and chaos in natural and manmade lumped and distributed sstems, Chemical Engineering Science 62, pp Fradkov, A.L. & Evans, R.J. 2005, Control of chaos: Methods and applications in engineering, Annual Reviews in Control 29, pp Glendinning P., 1994, Stabilit, Instabilit and Chaos, Cambridge Universit Press Goch, T.A.J. van, 2008, Simulation and validation of heat and airflow in an eperimental scale model. BSc thesis (In Dutch), Eindhoven Universit of Technolog; fac. Building and architecture, BPS, 106 pages Goch, T.A.J. van, Schijndel, A.W.M. van, 2008, Validation of DNS techniques for dnamic combined indoor air and constructions simulations using an eperimental scale model. European COMSOL Conference 2008 Hannover. (8 pages). Karatasou, S. & Santamouris, M., 2009, Derection of low dimensional chaos in building energ consumption time series. doi: /j.cnsns Lorenz, E. N. 1963, Deterministic nonperiodic flow, J. Atmos. Sci. 20 (2): Lu, W., Howarth, A.T., Jear, A.P., 1997, Prediction of airflow and temperature field in a room with convective heat source, Building and Environment 32(6) pp Schijndel, A.W.M. van, 2005, Implementation of FemLab in S-Functions, 1 ST FemLab Conference Frankfurt, pp Schijndel, A.W.M. van, 2007, Integrated heat air and moisture modeling and simulation, PhD thesis, Eindhoven Universit of Technolog Sinha, S.L., Arora, R.C. & Subhransu, R., 2000, Numerical simulation of two-dimensional room air flow with and without buoanc, Energ and Buildings 32, pp Tavazoei, M.S. & Haeri M., 2009, Chaos in the APFM nonlinear adaptive filter, Signal Processing 89, pp