# EE-489 Modeling Project: Behavior of the. Temperature Changes in a Room

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1 EE-489 Modeling Project: Behavior of the 1 Temperature Changes in a Room Younes Sangsefidi, Saleh Ziaieinejad, Tarik Wahidi, and Takouda Pidename, Abstract A precise heat flow model, which predicts the temperature changes in a building, is necessary in the design of buildings This report develops a heat flow medel to study the behavior of the temperature of a room when the window is opened First, a simple dynamic model for analyzing the air exchange of the room air with the outside, which is caused by a window opening, is studied Based on this model, a heat flow model is developed The developed heat flow model is implemented in MATLAB to study the temperature changes of the studied room The effectiveness of the proposed model and the presented simulation results are validated by an experimental setup using an ACURITE thermometer Based on the simulation and experimental results, some suggestions are discussed to improve the accuracy of the proposed model I INTRODUCTION Modeling and optimizing residential heat transfer is challenging and at the same time requires understanding of different physical models, their behavior and dynamics Research and experiments have been carried out in particular to understand the dynamics and save energy Heat loss or transfer from a room within a house or outside is well described by three mechanisms namely; conduction, convection and radiation It can be individual mechanism or combination of each In a room heat is transferred by conduction through solids such as windows, walls, roof and floor Heat lost by convection is leaking warm air to outside through windows and doors cracks and when they are open [1] A well-insulated room which means higher R-value of the material that the home is built from plays important role in transfer of heat The industry has characterized the R-value for different type of material In a room, windows offer least resistance to heat transfer and one third of the heat loss in winter occurs through windows [2] To better understand heat transfer through a window in a room, one must S Ziaeinejad, Y Sangsefidi, T Wahidi and T Pidename are with the School of Electrical Engineering and Computer Sience, Washington State University, Pullman, WA, (

2 develop an appropriate model that closely matches the physical characteristics of the room and the opening of window with respect to its area The behavior of the air flow to the room when a door is opened is studied in [3] Reference [3] shows that the air flows to the room in two phases: in the first phase, the constant air flow fills the room from the floor When the middle of the door is reached, the air flow continues in the second phase In the second phase, the air flow decreases exponentially until the upper level of the opening is reached Based on the model proposed in [3] for air flow when a door is opened, [4] proposes a model for the air flow to the room when a window is opened Reference [4] shows that the air flow to the room occurs in two phases: a constant air flow until the lower level of the opening is reached, and a decreasing air flow until the upper level of the opening is reached This report describes the model proposed in [4] for the air flow to the room when a window is opened, and shows that since that model does not consider the temperature changes, it cannot accurately model the air flow when the window is opened for a long time In addition, it cannot model the heat flow inside the room This report modifies the air flow model proposed in [4] and augments it with heat flow equations It will be shown that the result of this modification is coupled equations for modeling air flow and heat flow This report also presents the simulation results of the developed air flow and heat flow model, and validates the model and the simulation results by experimental results This report is organized as follows Section II describes the air flow model previously proposed in the literature Section III modifies the air flow model and augments the developed air flow model with a heat flow model Section IV presents the simulation and experimental results Section V discusses the results and suggests for improvement of the proposed model Section VI concludes the report II DESCRIPTION OF THE AIR FLOW MODEL PROPOSED IN [4] Reference [4] assumes that the opening is vertical, which is a sensible assumption for a window It assumes that there is no wind outside, and the mechanism for air flow into the room is the difference between the temperatures of the outdoor air and inside air The outside temperature is assumed to be constant and less than the temperature inside the room This is a valid assumption during the winter Reference [4] assumes that the window is open only for a short interval It is assumed that during this interval there is no heat transfer inside the room, and therefore the room temperature is constant Since it is assumed that there is no heat transfer, the temperature of the air flowing to the room is constant and equal to the outdoor temperature When the cold air flows into the room, it starts filling the room volume from the floor upwards At the same time, the warm air near the ceiling flows outside 2

3 the room Until the cold air level reaches to the lower level of the window, the air flow rate can be expressed as [5] B V in = C d g 3 H 0 3 (1) Where V in ( m3 s ) is the rate of the air volume flowing to the room, C d is the "opening orifice constant", which equals C d = (T in T out ), (2) Where T in is the temperature inside the room (K), and T out is the outdoor air temperature (K) In (1), B is the width of the opening and H 0 is the height of the opening g can be expressed as g = g 2(T in T out ) T in + T out, (3) where g is the acceleration due to gravity, and approximately equals 981 m s 2 After the cold air reaches to the lower level of the window (beginning of the second phase), the air flow rate can be expressed as where H(t) equals B V in (t) = C d g H(t) 3, (4) 3 H(t) = 1 ( at ) 2, (5) H0 where a equals B B a = C d g = C d g 2(T in T out ), (6) 3A g 3A g T in + T out where A g is the room floor area As the time passes, H(t) becomes smaller and V in (t) decreases 3

4 Fig 1 n layers of the room air III DEVELOPING A COUPLED HEAT AND AIRFLOW MODEL The air flow model described in Section II is valid only if the window is open for a short interval In case that the window is opened for a long time, the assumption of a constant room temperature is not valid In addition, we can not assume that the air that has entered the room will stay near the floor for ever Therefore, it is needed to augment the air flow model with a heat model This results coupled equations Since the heat flow leads to movements of the air molecules, it is impossible to find the location of the air which has entered the room Therefore, it is impossible to distinguish between different phases of the air flow For simplicity, it is assumed that during the entire process, the air flow happens in phase 1 There is no assumption of constant room temperature It is assumed that except for the open window, there is no heat transfer between the air inside the room and the outside All the walls and doors are assumed to be ideally insulated It is assumed that different points with the same height have the same temperature As shown in Fig 1, we can partition the room into n layers, all with the same thickness It is assumed that the incoming cold air goes below the first layer, and affects the temperature of the first layer Because of the differences in the air density, l 1 is the coldest layer, and l n is the warmest When the outdoor air enters the room, it is replaced with a volume of the n th layer air Therefore, the rate of the heat transferred to l 1 from the outdoor air is Q in = ṁ in (t)c air (T OUT T n ) = V in (t)ρ air C air (T OUT T n ) (7) In (7), ρ air and C air are the density and specific heat capacity of the air T n is the temperature of the last layer, and V in (t) equals V in (t) = (T 1 T out ) B 3 4 2g (T in T out ) T in + T out H(t) 3 (8)

5 The rate of the heat transferred to each layer can be found by subtracting the rate of the heat coming from the bottom layer and the rate of the heat going to the upper layer Q 1 (t) = Q in (t) Q 12 (t) Q 2 (t) = Q 12 (t) Q 23 (t) Q 3 (t) = Q 23 (t) Q 34 (t) (9) Q n 1 (t) = Q (n 2)(n 1) (t) Q (n 1)n (t) Q n (t) = Q (n 1)(n) (t) Assuming l i and l i+1 are two adjacent layers, convection, which can be expressed by Q i(i+1) (t) is the convective heat transferred by the Q i(i+1) (t) = h c A g (T i T i+1 ) (10) where h c is the convective heat transfer coefficient After finding the rate of the heat transferred to each layer, the rate of the change in the temperature of that layer is T i (t) = where V l is the volume of each layer, and equals Q i (t) ρ air V l C a ir (11) V l = V room n = L roomw room H room n where L room, W room, and H room are length, width, and height of the room, respectively The behavior of the room temperature after opening the window can be simulated by the Euler method, as described below 1) T 1, T 2,,T n are considered as state variables The initial values of all the state variables are equal to the steady-state room temperature before the window is opened (12) T 0 1 = T 0 2 = = T 0 n = T 0 in,steady state (13) 5

6 2) The sampling time for simulation is chosen t At the k th sampling time, Q k in is calculated from (7) Then, the temperature vector [ T k+1 1 T k+1 1 T k+1 n ] T (14) can be found by T k+1 1 T k+1 2 T k+1 n = T k 1 T k 2 T k n + T k 1 T k 2 T k n t (15) where T k 1 T k 2 T k 3 T k n 1 T k n = h ca g ρv l C air T k 1 T k 2 T k 3 T k n 1 T k n + 1 ρv l C air Q k in (16) Equation (16) is nonlinear because Q k in is a nonlinear function of the states and the input of the system (B) The output of the system y k equals y k = selector 1 n T k 1 T k 2 T k n (17) where selector 1 n is a 1 n matrix Except one entry of selector which is 1, the other entries are zero To form selector 1 n, we should find the layer that inclused our point The entry of selector 1 n 6

7 TABLE I DIMENTIONS AND THE PARAMETERS OF THE SYSTEM Room length, L room Room width, W room Room length, H room Window width, B Window length, H 0 Window height from the ground, H w Test point height from the ground, H point 338 m 310 m 240 m 0272 m or 0544 m 1126 m 1 m 1 m Convective heat transfer coefficient, h c 5 Specific heat capacity of the air, C air Density of the air, ρ air Outside temperature, T out Joule m 2 s C 1000 Joule kg C 129 kg m 3 8 C Room temperature before opening the window, T in,0 267 C which corresponds to that layer is chosen 1 the other entries are chosen 0 After finding y 1,y 2,,y k,, we can plot y (the temperature of our selected point) as a function of time IV SIMULATION AND EXPERIMENTAL RESULTS To study the performance of the proposed temperature model, the model is simulated using MATLAB The parameters of the system are shown in Table I In the first simulation case study, the window is opened 0272 m In this case, the room is devided to 5 layers Fig 2(a) shows the volume of the air entered the room from the outside It can be seen that when the difference between the temperatures of the room and outside decreases, the air flow rate decreases Fig 2(b) shows the temperature of 5 different layers As expected, the temperature of the bottom layers decreases faster The reason is that the cold air stays at the bottom of the room Fig 3 shows the simulation results of the temperature changes when the window is opened 0272 m and the number of layers is defined 10 By increasing the number of layers, it is easier to predict the temperature of different points in the room with higher accuracy In this case study, lower layers still have faster decreases in their temperatures Fig 4 shows the simulation results of the temperature changes when the width of the window opening is set to 0544 m The number of layers is defined 10 It can be observed that a larger opening leads to faster decrease in the temperatures of all layers Figs 2, 3, and 4 show the first few minutes of the experiment, which is good for observing the differences of the temperatures of different layers at the beginning of the experiment The temperatures of all layers finally converge to T out Since the doors 7

8 Fig 2 Simulation results for B = 0272 m; (a) volume of the air entered the room from outside and (b) temperature of five different layers of the room Fig 3 Simulation results of the temperature of ten different layers of the room for B = 0272 m and walls are assumed to be ideally insulated, it is expected that the temperatures of all the points inside the room eventually become equal to T out To validate the proposed model, an experiment is performed and the temperature is monitored using an ACURITE thermometer, which is shown in Fig 5 The parameters of the experiment are similar to the parameters listed in Table I The width of the window opening is B = 0272 m and the thermometer sensor is mounted at the height of 082 m Fig 6 shows the experimental result It can be seen that in the first minutes of the test, there is a good agreement between the simulation and 8

9 Fig 4 Simulation results of the temperature of ten different layers of the room for B = 0544 m Fig 5 Configuration of the test setup experimental results In the first few minutes of the experiment, the difference between the room temperature and the temperatures of different parts of the house is small Therefore, the heat transfer from the walls and the door is negligible, and the model assumption (perfect insulation) is almost valid The agreement between the simulation and experimental results during this period shows the effectiveness of the proposed model However, at the end of the studied period, the difference between simulation and experimental results becomes more significant Although we tried to minimize the heat transfer between the room and other parts of the house, this heat transfer is not zero and it becomes higher when the room temperature becomes much less than the temperature of other parts of the house A precise model should also consider the heat transfers between the room and other parts of the house 9

10 Fig 6 Experimental and simulation results of the temperature of the studied point at the height of 0817 m when B = 0544 m V DISCUSSIONS AND MODEL IMPROVEMENT Based on the discussed model and presented results, there are some suggestion for increasing the accuracy of the model: 1) As previously discussed, there is still a heat transfer between the room and other parts of the house A precise model should also consider the heat transfer through the walls, door, ceiling, floor, and available openings in the room 2) In the proposed model, the density of the air is assumed constant However, it depends on the air temperature and slightly varies with the variation of the temperature It is suggested to consider an adaptive density model based on the temperature 3) In the presented model, the temperature of the points at the same layer are equal However, the points near the window are colder than the points with the same height but far from the window This issue can be considered in the improvement of the model It is possible to divide the room volume into different cubes which transfer the heat with the adjacent cubes from each of their 6 sides This model is more complicated but more accurate VI CONCLUSION This report studies the heat flow between a room and the outside when a window is opened Based on the studied heat flow model, this report presents a method to calculate the profile of the temperatures of different layers of the room The presented model can be defined for different number of layers The effectiveness of the proposed model is validated using simulation and experimental results Based on the results, in a room with an opened window, the decrease in the temperature of the bottom layers 10

11 is more than the decrease in the temperature of the top layers In addition, since there are some heat leakages in the practical case study, after fast transients, the room temperature in experimental case study is higher than the temperature of the simulation case study If the heat flow happens only through the opened window, the model is more precise to predict the temperature Based on the simulation and experimental results, some suggestions are discussed to improve the accuracy of the proposed model VII APPENDIX (MATLAB CODE) The proposed model for analysis of temperature changes is implemented in MATLAB Fig 7 shows the MATLAB code REFERENCES [1] M Rubin, Calculating heat transfer through windows, International Journal of Energy Research, vol 6, no 4, pp , 1982 [2] J F Kreider, P Curtiss, and A Rabl, Heating and cooling of buildings: design for efficiency McGraw-Hill New York, 1994 [3] D Kiel and D Wilson, Gravity driven flows through open doors, in 7th AIVC Conference, 1986 [4] B Nordquist and L Jensen, A dynamic model for single sided ventilation Air Distribution in Rooms: Ventilation for Health and Sustainable Environment, p 211, 2000 [5] G Gan, H Awbi, and D Croome, Simulation of air flow in naturally ventilated buildings, Proceedings of Building Simulation, Nice, France, pp 78 84,

12 Fig 7 MATLAB code 12

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