Simulation and Experimental Validation of Chaotic Behavior of Airflow

Volume 2

9th Nordic Symposium on Building Physics - NSB 2011

Simulation and Experimental Validation of Chaotic Behavior of Airflow in a Ventilated Room Jos van Schijndel, Assistant Professor Eindhoven University of Technology, Netherlands KEYWORDS: Airflow, chaos, system, ventilation SUMMARY: Chaos may lead to instability, extreme sensitivity and performance reduction in dynamic systems. Therefore it is unwanted in many cases. Due to these undesirable characteristics of chaos in practical systems, it is important to recognize such a chaotic behavior. In this paper the chaotic behavior of the airflow in case of an ordinary ventilated room is researched. Computational chaotic behavior is already observed in the simulations by changing the supply air temperature from 22 oC into 21.9 oC However, it could be the case that, despite all efforts, the chaotic behavior is a numeral artifact. Therefore laboratory experiments using a scale model were performed. It is concluded that the computational model of the experimental scale model has to be improved to simulate the experimental results more accurately.Furthermore, the presented method seems promising in detecting chaotic behaviour.

1. Introduction Chaos theory is a field of study in mathematics, physics, economics, and philosophy studying the behavior of dynamic systems that are highly sensitive to initial conditions. This sensitivity is popularly referred to as the butterfly effect. Small differences in initial conditions (such as those due to rounding errors in numerical computation) yield widely diverging outcomes for chaotic systems, rendering long-term prediction impossible in general. This happens even though these systems are deterministic, meaning that their future behavior is fully determined by their initial conditions, with no random elements involved. In other words, the deterministic nature of these systems does not make them predictable. This behavior is known as deterministic chaos, or simply chaos. Chaotic behavior can be observed in many natural systems, such as the weather. Explanation of such behavior may be sought through analysis of a chaotic mathematical model. Two icons of the chaos theory 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

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Figure 1 left, shows the bifurcation diagram of the logistic map xn+1 → r xn (1 – xn). This map represents a discrete model for a dynamic system. Each vertical slice shows the attractor (solution of x) for a specific value of r. The diagram displays period-doubling as r increases, eventually producing chaos in the dynamic system. Figure 1 right, presents the Lorenz attractor that also displays chaotic behavior in a dynamic system. These two plots demonstrate sensitive dependence on initial conditions within the region of the phase space occupied by the attractors. Moreover, chaotic systems may lead to instability, extreme sensitivity and performance reduction. Therefore it is unwanted in many cases. Due to these undesirable characteristics of chaos in practical systems, it is important to recognize such a chaotic behavior. The existence of chaos has been discovered in several areas during the last 30 years. However, there is a lack of studies in relation with buildings that also can be regarded as complex dynamic systems as well. Furthermore, chaotic behavior may have an enormous impact on the predictability of the indoor climate. For example, if the indoor climate has, under certain circumstances, a similar dynamic behavior as the Lorenz attractor then it could spontaneously and without any detectable cause turn from hot into cold. In an extreme situation this may lead to fully 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 ordinary ventilated room with an on/off controller. The approach to detect chaos in a given system is to investigate both numerical as well as experimental results. There are several universal indicators for chaos. In this research we use the property that all chaotic systems are extreme sensitive for initial conditions and/or system parameters, i.e. small differences in them can lead to extraordinary differences in the system states. Our research methodology 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 study and reproduce simulation results; (3) Investigate computational chaotic behavior using this case study by small changes in system parameters. If promising results are obtained, proceed with: (4) Experiments in a scale model. The paper is organized in the same way.

2. Numerical case study: Airflow in a ventilated room A case study was selected based on the work of Sinha et al. (2000). The subject of this case study comprehends airflow in a ventilated room. Figure 2 shows the geometry.

FIG 2. The geometry of the airflow problem. The PDEs are based on the well-known Navier-Stokes equations :

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u t

(uu) x

(vu) y

p x

v t

(uv) x

(vv) y

p y

T t

(uT ) x

(vT ) y

1 Re

2

1 Re

1 Re Pr

2

2

u

v

Gr u T 2 x Re

v y

0

T

The boundary conditions are: At the left, right, top and bottom walls: u=0, v=0, T=0. At the inlet:

u=1, v=0, T=1.

At the outlet :

Neuman conditions for u,v and T

The temperature solutions for several Re and Gr numbers from Sinha et al. (2000) are shown:.

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 buoyancy is taken into account (i.e. Gr = 2.5 107 and Re = 1000).

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Figure 4 shows the temperature distributions after respectively 20 and 40 time steps in case of the constant inlet air temperature boundary condition.

FIG 4. Complex airflow patterns after 20 (left) and 40 (right) time steps.

3. Chaotic behavior using the numerical case study In order to detect chaotic behavior, the airflow was again simulated with the only difference that the supply air temperature was slightly changed from 22 oC into 21.9 oC (i.e. 1 into 0.975 for the scaled temperature). In Figure 5, the results are presented.

FIG 5. Scaled temperature distribution after 60 time steps (case: Gr = 2.5 107 and Re = 1000); Top Left: Air supply temperature equals 1 (22 oC); Top Right: Air supply temperature equals 0.975 (21.9 o C); Bottom: Air temperature difference between the top figures where 0 means no difference; +1 means 4 oC hotter; -1 means 4 oC colder

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Figure 5 shows the temperature distribution after 60 time steps. The top left figure has an air supply temperature of 22 oC and the an air supply temperature of the top right figure, 21.9 oC 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 very sensitive for the air supply temperature: A small change of 0.1 oC in the air supply temperature (i.e. 1 into 0.975 for the scaled temperature) may temporary and locally 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 (Relay) 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 Relay provides input for this SimuLink block and is located at position x=2; y=2.5 in the room represented by Figure 2. The Relay switches the supply temperature (scaled) between 1 (hot, 22 oC) and 0 (cold 18 oC) if the ‘sensor’ temperature is respectively below 0.3 (i.e. 19.2 oC ) and above 0.5 ( i.e. 20 oC). In order to investigate the occurrence of chaotic behavior analog to the beginning of this Section, a slight change of a system 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 experiment are comparable with Figure 5. As expected, this produces even greater differences between the two simulations, with just one slightly different parameter setting. The numerical experiments presented in this Section provide evidence for chaotic behavior, such as small differences in parameters yield widely 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 experiments with a scale model.

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4. Experimental case study: A scale model 4.1

Measurements

Figure 7 shows the experimental scale model, with external dimensions 0.64m x 0.45m x 0.39m (van Goch et al. 2008). Each length scale (i.e. x, y, z) is a factor 7.7 smaller than the well documented full scale room experiment of Lu et al. (1997) The dimensions are shown in Figure 7.

FIG 7. The experimental scale model. The sides and top of the box are constructed as follows: 3 mm Plexiglas – 4 mm air cavity – 3 mm Plexiglas. The material properties of Plexiglas are: heat conductivity are: 0.21 W/mK; density, 1190 kg/m3; heat capacity, 1500 J/kgK. The window and bottom are made of 4 and 6 mm Plexiglas respectively. Three different types of measurements were completed: thermal imaging, air flow visualization using smoke, acquisition of temperature sensors over time. Thermal images were obtained of the external side including the windows (front). These images are very useful for validation purposes because they 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 box. 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 typical result obtained during steady state (after 120 min and 30W input power) conditions. The first step was to reproduce these experimental results numerically (see next section). The second step would be to study whether small changes in the experiment would lead to large effects.

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4.2


Computational model

The modeling details are published in van Goch et al. (2008). A short summary is presented below. The combination of the ‘Convection and Conduction’ and ‘Navier-Stokes’ application modes of Comsol are used to model and simulate the experiments. Furthermore only default grids (course, fine, etc.) and solvers are used. The calculations were limited to a maximum of 32 GB memory 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 velocity after 900 seconds. Figure 10 (right) provides the air circulation

FIG 9. Left: Right: Simulated surface temperature

FIG 10. Left: The temperature and velocity after 900 seconds; Right: The air circulation After evaluation of all results it has been determined the primary reason for the discrepancy between the results obtained from the simulation and those of the experiments on the model enclosure was most likely due to the modeling of the heat source. The simulated air velocity is simply too low just above the heating coil. To achieve the desired agreement between results of the experimental work and that of the simulation, it is likely better to model the heat source in greater detail (to improve the simulated velocity just above the heat source) or use a velocity profile instead of natural convection. Evidently, the computational model needs improvement if it is to accurately simulate the experimental results; this is the focus of current and on-going research. After this validation our aim is to study the effect of small changes on both the experimental scale model as well as the computational model.

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5. Conclusions Chaos may lead to instability, extreme sensitivity and performance reduction in dynamic systems. Due to these undesirable characteristics of chaos in practical systems, it is important to recognize such a chaotic behavior. The existence of chaos has been discovered in several areas during the last 30 years. However, there is a lack of studies in relation with buildings that also can be regarded as complex dynamic systems. In this paper the chaotic behavior of the airflow in case of an ordinary ventilated room is researched. Computational chaotic behavior is already observed in the simulations by changing the supply air temperature from 22 oC into 21.9 oC However, it could be the case that, despite all efforts, the chaotic behavior is a numeral artifact. Therefore laboratory experiments using a scale model were performed. The first step was to reproduce these experimental results numerically. We conclude that the computational model has to be improved to simulate the experimental results more accurately. The second step will be to study whether small changes in the experiment will lead to large effects on both the experimental 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 expected so far.

References Cai, W., Sen, M., Yang, K.T. & McClain, R.L., 2006, Synchronization of self-sustained thermostatic oscillations in a thermal-hydraulic network, Int. Journal of Heat and Mass Transfer 49, pp44444453 Elnashaie, S.S.E.H. & Grace, J.R., 2007, Complexity, biconfiguration and chaos in natural and manmade lumped and distributed systems, Chemical Engineering Science 62, pp3295-3325 Fradkov, A.L. & Evans, R.J. 2005, Control of chaos: Methods and applications in engineering, Annual Reviews in Control 29, pp33-56. Glendinning P., 1994, Stability, Instability and Chaos, Cambridge University Press Goch, T.A.J. van, 2008, Simulation and validation of heat and airflow in an experimental scale model. BSc thesis (In Dutch), Eindhoven University of Technology; fac. Building and architecture, BPS, 106 pages Goch, T.A.J. van, Schijndel, A.W.M. van, 2008, Validation of DNS techniques for dynamic combined indoor air and constructions simulations using an experimental scale model. European COMSOL Conference 2008 Hannover. (8 pages). Karatasou, S. & Santamouris, M., 2009, Derection of low dimensional chaos in building energy consumption time series. doi: 10.1016/j.cnsns.2009.06.022 Lorenz, E. N. 1963, Deterministic nonperiodic flow, J. Atmos. Sci. 20 (2): 130–141 Lu, W., Howarth, A.T., Jeary, A.P., 1997, Prediction of airflow and temperature field in a room with convective heat source, Building and Environment 32(6) pp541-550 Schijndel, A.W.M. van, 2005, Implementation of FemLab in S-Functions, 1ST FemLab Conference Frankfurt, pp324-329. Schijndel, A.W.M. van, 2007, Integrated heat air and moisture modeling and simulation, PhD thesis, Eindhoven University of Technology Sinha, S.L., Arora, R.C. & Subhransu, R., 2000, Numerical simulation of two-dimensional room air flow with and without buoyancy, Energy and Buildings 32, pp121-129 Tavazoei, M.S. & Haeri M., 2009, Chaos in the APFM nonlinear adaptive filter, Signal Processing 89, pp697-702

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