ABSTRACT INTRODUCTION

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1 A modelling of indoor air flow and heat transfer through door and window by free convection Guang-Fa Yao, J.A. Khan Department of Mechanical Engineering, University of South Carolina, Columbia, South Carolina 29208, USA ABSTRACT In the present work, an investigation of natural convection in partitioned enclosure is made. The effects of following parameters on convection transport were studied; (1) Enclosure aspect ratios A^=L/H; (2) Aperture ratios Ap=h/H; (3) yleigh numbers. Parametric study is made for two hundred different cases. This work is significantly different from similar numerical works because we studied the flow at higher yleigh numbers than were studied previously. Both the heat transfer and fluid exchange rates through the opening in the partition are studied. The knowledge of volume exchange and heat transfer rates can be directed toward modelling the contaminant transport in buildings and indoor air quality study. INTRODUCTION In order to investigate indoor air pollution and indoor air quality, the fluid flow and the heat transfer in the form of natural convection through doors or windows have been simulated as natural and forced convection in enclosures with partition or partial opening. Some of the experimental and numerical works done in this area is described below. Brown and Solvason [1] and Brown [2] conducted experimental investigations of natural convection heat and mass transfer through a small opening in a vertical partition and a horizontal partition between two chambers. A theoretical correlation for air was derived using Bernoulli's principle. The experimental results were then used to validate the theory. Shaw [3] developed a theory to calculated volumetric exchange of air due to natural convection and combined natural convection and forced flow through a rectangular opening in a vertical partition. Epstein [4] and Epstein and Kenton [5] performed experiments dealing with buoyancy-driven exchange

2 196 Air Pollution flow and combined natural convection and forced flow through small openings in horizontal and vertical partitions. In these experiments, brine solution was used to model air transport, viscous and temperature effects appear to be neglected. Bajorek and Lioyd [6] made an experimental investigation of natural convection in partitioned square box with air and carbon dioxide as the working fluid. Similar model was studied by Zimmerman and Acharya [7] numerically. Acharya et al [8] also made another numerical investigation where a poorly conducting partition protrudes upwards from the floor of enclosure. Both partition and aperture ratios effects on heat transfer was explored at low yleigh number(< 10*). Both of these work were performed using a control volume based finite difference method SIMPLE. Shaw et al [9] performed similar numerical research with perfectly adiabatic partition, but used a cubic spline collocation method. Chang et al [10] made a numerical (finite element) study of square box with partition projecting downwards from the ceiling of enclosure. This work deals with the effects of partition and yleigh numbers(< 10*) in a transient state. Nansteel and Grief [11] set up an experiment which dealt with water filled enclosure(enclosure aspect ratio=2) at very high yleigh numbers 2.3X10** to 1.1X10** based on enclosure length. The effect of the partition on the fluid flow and temperature fields was investigated by dye-injection flow visualization and by thermocouple probe, respectively. A similar experimental work done by Lin and Bejan [12] deal with yleigh number range of 10* to 10* based on enclosure height and for aperture ratios, 1, 1/4, 1/8, 1/16 and 0. Nansteel and Greif [13] also carried out experimental studies of natural convection in a water-filled enclosure with an aspect ratio (H/L) of 0.5. The effect of the transverse location and the upward or downward extension of division were examined. Neymark et al [14] carried out an experimental research of natural convection in a cube geometry, with a vertical partition in the center. The convective transport in air and water were compared. Recent works in this area include Karki et al [15], Sathyamurthy et al [16] and Hawkins et al., [17]. PHYSICAL MODEL The three different geometries studied along with their boundary conditions are shown in Figure 1. The two horizontal walls are considered insulated while the two vertical walls are maintained at two different temperatures T% and T, respectively. The partition is assumed to be insulated. The parametric study includes investigations for aspect ratios of (A^=L/H) 1 and 2 and aperture ratios of (A,=h/H) 1/2, 1/4, 1/6, 1/8, 1/10. The flow is simulated as a two dimensional phenomenon with the assumptions that the fluid is incompressible and the flow is laminar. diation effects are neglected and Boussinesq approximation is invoked. Introducing the following dimensionless variables;

3 y=-z H' H' Air Pollution 197 (1) ocv The governing differential equations that express the conservation of mass, momentum and energy in the fluid domain become: BU+dV^ (2) dx* BY dy dx dx* 8Y* (3) dx dy dy (4) y^e+k = ( + ) (5) 8X BY Pr dx* BY* the non-dimensional Nusselt number, Nu, and volumetric exchange flow rate, Q, are defined as in equation 6. <?-/** UdY ar (6)

4 198 Air Pollution NUMERICAL PROCEDURE Equations 2 through 5 are discretized utilizing a control volume based finite difference method. The presence of solid area is accounted for by a strategy developed by Patankar [18] in which the governing equations along with the boundary conditions are solved for the complete domain with the solid area characterized by a very high (10^ ) viscosity and zero conductivity. A SIMPLE-like algorithm developed by Date, [19], is used to treat the coupling of momentum and energy equations. In this method, at every iteration level, the solution of discretization equations consist of two stages, prediction stage and correction stage. Details of the numerical procedure is documented elsewhere [17,18,19]. The following convergence criterion was adopted: ' «' ' Where ej and e^ are 10"* and 10"* respectively, $ indicates the physical variable of interest, and superscript k denotes the iteration index. RESULTS AND DISCUSSIONS In the present investigation, the yleigh numbers are varied from 10* to 5X10^ and the Prandtl number used is 0.71 which corresponds to air. Two enclosure aspect ratios (1 and 2) and five aperture ratios (1/2, 1/4, 1/6, 1/8, 1/10) are studied. Heat transfer and fluid flow through the partition are analyzed. The measure of volume exchange rate is an indicator of contaminant transport through doors and windows in a building. Fluid Fields The velocity vectors for an aspect ratio (A^,=H/L) of one, and for an aperture ratio (Ap = h/h) of 1/10 are presented in Figure 2. At low yleigh numbers(l(x* to 10*), fluid is heated along the hot wall and is driven upward by buoyancy effect. For geometries (a) and (c) a clockwise circulation is formed on both sides of the partition. These cells are more pronounced and are similar on both sides of the partition at smaller aperture ratios, as shown in Figure 2 (a), (b), (e) and (f). At higher Ap=l/2 and >10\ the fluid flow has the characteristic of a boundary layer flow. This characteristics is more pronounced at high yleigh number of =5X10\ ( this case is not shown in the figure). At this yleigh number the fluid separates from the

5 Air Pollution 199 main stream at both upper left corner and the bottom left corner in the hot chamber. The main fluid stream separates from cold wall at an elevation corresponding to the partition height (h), and then flows horizontally toward the hot chamber. Boundary layer thickness in both the hot wall and cold wall is observed to decrease dramatically with the increase in yleigh number. At Ap< 1/4, a stronger fluid flow was found to exist in the hot chamber for the case where the partition projects upward ( Figure 2(b)) than in cold chamber, whereas, a stronger field is observed in the cold chamber when the partition projects downward (Figure 2(0). The velocity vector for the fluid flow corresponding to geometry (b) are shown in figures 2(c) and (d), at higher yleigh numbers two clockwise circulation cells are created on both sides of the partition. Also, at yleigh numbers larger than 10*, the fluid flow has a strong characteristics of boundary layer flow. With the increase of yleigh number, fluid entrapment phenomenon occurs both at the top area of hot chambers and the bottom area of cold chambers. When aperture ratios decrease, two stream, hot stream going toward cold chamber, and cold stream approaching hot chamber, contact at the opening of the partition. Another impressive fluid flow feature for this geometry is that fluid fields are strictly symmetric to the partition planes. This is consistent with what will be expected according to the governing equations and boundary and geometry conditions. The fluid flow field for the enclosure aspect ratio of 2 has similar trend, and is not discussed in this paper. Temperature Fields The temperature distribution in a room is important during the investigation of indoor air quality. For example, in the work done by Sathyamurthy et al[16], the contaminant concentration at a monitoring location was defined as a function of dimensionless temperature. Therefore, the temperature distribution can be translated into the contaminant concentration distribution in enclosures. Figure 3 presents the isotherms in the enclosure at an aspect ratio of one and an aperture ratio of 1/2, at two different yleigh numbers. The temperature profiles for enclosure aspect ratio of two is very similar to that for aspect ratio of one, therefore, they are not shown here. By studying the isotherms the following characteristics are observed: (1) at high yleigh numbers (Figure 3(b), 3(d) and 3(e)), the fluid in the enclosure is found to have very strong thermal stratification; and (2) at small aperture ratios (not shown in the Figure), very dense isotherms at the opening area of partition is observed. This means that the temperature gradients in this region are substantial. These strong temperature gradients may explain why we obtained strong air circulation in the hot chamber for Geometry (a) and in the cold chamber for Geometry (c). The Heat Transfer te Figure 4 presents the average Nusselt numbers for two enclosure aspect ratios and for all the aperture ratios studied for all three geometries studied in

6 200 Air Pollution this work. Comparison of Figure 4 (a), (b) and (c) which are for aspect ratio of 1, with Figure 4 (d), (e) and (f) which are for aspect ratio of 2, reveal that the effects of yleigh numbers on heat transfer is very similar at both the enclosure aspect ratios. The effects of yleigh numbers on the Nusselt numbers are stronger at higher yleigh numbers. This is expected because at low yleigh numbers, the exchange flow rate through opening is negligible and heat conduction between the two chambers at the opening plane dominates the heat transfer mechanisms. Although the Nu~ curves for A^ = l are similar to the those for A^=2, the magnitude of the Nusselt numbers A^, = l are 0 to 23% larger than those at A^=2, depending on the aperture ratios and yleigh numbers. Also, for the same enclosure aspect ratios, the Nusselt numbers corresponding Geometry (b) are greater than those for Geometry (a) and (c). For A^ = l, the Nusselt numbers for Geometry (b) are 11.5 to % larger than those for Geometry (a) and (c). Therefore, it is obvious that an opening in the middle of the partition allows maximum heat transfer. Volume Exchange Flow te Figures 5 and 6 show volume exchange flow rates as a function of yleigh number and aperture ratios. The yleigh number has a greater effect on volume exchange flow rates at low yleigh numbers(<10*) than at higher yleigh numbers. But as the exchange flow rate at low yleigh numbers are negligible, this fact is not significant. It is interesting to note that, for Geometry (b), (both at A^=l and A^=2) at higher yleigh numbers > 10\ when aperture ratio decrease, exchange flow rate increases, as shown in Figure 6. Comparing the three geometries it was found that the exchange flow rate for geometry (b) was.2 to 210% higher than the corresponding cases for geometry (a) and (c). CONCLUSION AND SUGGESTION FOR FUTURE WORK In this work, a detailed numerical investigation of flow in partitioned enclosures was made. About two hundred cases were studied using SIMPLE and a modified SIMPLE like method to deal with the coupling among the governing equations. The following conclusion are made: 1. Both yleigh numbers and aperture ratios have strong effects on heat transfer and fluid flow. With A,< 1/4, aperture ratios has stronger effect on both the Nusselt numbers and exchange flow rates. However, at yleigh numbers larger than 10^ this effect decrease sharply; except for the cases which corresponds to geometry l(b). 2. With the increase of enclosure aspect ratios the Nusselt numbers decrease. The Nusselt numbers for A^= 1 are 0 to 22% larger than those with Ap=2 for Geometry l(a) and (c), 0 to 22.4% for Geometry l(b), depending on yleigh numbers and aperture ratios. As for the volume exchange flow

7 Air Pollution 201 rates, when enclosure aspect ratio increases the exchange flow rates decreases at low yleigh number(< 10"), whereas, the exchange flow rate increases at higher yleigh numbers. 3. Comparing three geometries, the geometry where there is opening in the middle of the partition (Figure l(b)) allows the maximum heat transfer. Also, for this geometry a maximum fluid exchange flow rate was found at about Ap= 1/4 for = 10^. This fact has practical importance for the better design of ventilation systems. 4. With the decrease of aperture ratios, stronger air circulations are observed in the hot chambers for Geometry (a) and cold chambers for Geometry (c). At higher yleigh numbers, some fluid was trapped at the bottom of the cold chamber for Geometry (a), at the top of hot chamber for Geometry (c), and both at the top of hot chamber and the bottom in cold chamber for Geometry (b). 5. The temperature distribution at higher yleigh numbers has strong thermal stratifications. When the aperture ratios decrease, higher temperature gradients are found in opening area, which is partially responsible for the stronger air circulation in the hot chambers for Geometry (a) and cold chambers for Geometry (c) as aperture ratios decrease. REFERENCE [1] W. G. Brown and K.R.Solvason, Natural Convection Heat Transfer Through Rectangular Openings in Partitions-1, Int. J. Heat Mass Transfer Vol. 5, pp (1962). [2] W.G.Brown, Natural Convection Through Rectangular Openings in Partitions-2, Int. J. Heat Mass Transfer, vol.5, pp (1962). [3] B. H. Shaw, Heat and Mass Transfer by Natural Convection and Combined Natural Convection and Forced Air Flow through Large Rectangular Openings In Vertical Partitions, Symposium on Heat and Mass Transfer by Combined Forced and Natural Convection, IMechE, Paper No. Cl 17/71, [4] M.Epstein, Buoyancy-driven Exchange Flow Through Small Openings in Horizontal Partitions, ASME J. Heat Transfer, Vol. 110, pp (1988). [5] M. Epstein and M. A. Kenton, Combined Natural Convection and Forced Flow Through Small Openings in a Horizontal Partition, with Special Reference to Flows in Multicompartment Enclosures, ASME J. Heat Transfer, Vol. Ill, pp (1989). [6] S. M. Bajorek and Liyod, J. R., Heat Transfer Due to Buoyancy in a Partially Divided Square Box, Int. J. Heat & Mass Transfer, Vol. 104, pp , 1982.

8 202 Air Pollution [7] E.Zimmerman and S.Acharya, Free Convection Heat Transfer in a Partially Divided Vertical Enclosure With Conducting End Walls, Int. J. Heat Mass Transfer, Vol. 30, No. 2, pp (1987). [8] S.Acharya and R.Jetli, Heat Transfer Due to Buoyancy in a Partially Divided Square Box, Int. J. Heat Mass Transfer, Vol. 33, No.5, pp (1990). [9] H.J.Shaw, C.K. Chen and J.W. Cleaver, Cubic Spline Numerical Solution for Two Dimensional Natural Convection in a Partially Divided Enclosure, Numerical Heat Transfer, Vol.12, pp (1987). [10] L.C. Chang, J.R. Lioyd and K.T. Yang, A Finite Difference Study of Natural Convection in Complex Enclosure, Proc. 7th. Int. Heat Transfer Conf., Munich, Federal Republic of Germany, pp (1982). [11] M.W.Nansteel and R. Greif, Natural Convection in Undivided and Partially Divided Rectangular Enclosures, ASMEJ Heat Transfer, Vol. 103, pp (1981). [12] N. Lin and A. Bejan, Natural Convection in a Partially Divided Enclosure, Int. J. Heat & Mass Transfer, Vol.26, No. 12, pp (1983). [13] M.W.Nansteel and R.Greif, An Investigation of Natural Convection in Enclosures with Two and Three-dimensional Partitions, Int. J. Heat & Mass Transfer, Vol.27, No.4, pp (1984). [14] J. Neymark, C.R. Board man and A. Kirkpatrick, High yleigh Number Natural Convection in Partially Divided Air and Water Filled Enclosure, Int. J. Heat & Mass Transfer, Vol.32, No.9, pp (1989). [15] K. C. Karki, P. S. Sathyamurthy and S. V. PatankarP. S., Natural Convection in a Partitioned Cubic Enclosure, ASME Journal of Heat Transfer, Vol. 114, pp , [16] P. S. Sathyamurthy, K. C. Karki, and S. V. Patankar,K. C., Laminar Mixed Convection in a Partitioned Enclosure, Natural Convection in Enclosures, ASME, HTD-Vol. 198, p. 57, [17] L. E. Hawkins, J. A. Khan, and G. Yao, A Numerical Solution of Buoyancy-Driven Flow through Small Openings Between Two Enclosures, Natural Convection in Enclosures, ASME, HTD-Vol. 198, pp , [18] S. V. Patankar, A Numerical Method for Conduction in Composite Materials, Flow in Irregular Geometries and Conjugate Heat Transfer, Proc. Sixth Int. Heat Transfer Conf., Toronto, Vol. 3, p 297, [19] A. W. Date, Numerical Prediction of Natural Convection Heat Transfer in Horizontal Annulus, Int. J. Heat and Mass Transfer, Vol. 29, No. 10, pp (1986).

9 Air Pollution 203 T, h (a) (b) s T, (c) Figure 1. Geometry and Boundary Conditions

10 204 Air Pollution (a) (b) (c) (d) (e) (0 Figure 2. Velocity Vectors for L/H= 1 (a), (c) and (e): h/h=l/10, =lx10«; (b), (d) and (f): h/h=l/10, =5X10';

11 Air Pollution 205 (a) (b) (c) (d) (e) Figure 3. Isotherms for L/H= 1 (a), (c) and (e): h/h=l/2, =lx10*; (b), (d) and (f): h/h=l/2, =5X10*; (0

12 Air Pollution : 10: 1 : 0.1 '10' 10* 10*10* 10' 10 ' (a) 100i 100i 10 * 10 * ' 10 ' (b) 10: 10: 1 : 100 '10 * 16 L 16 '10 (c) 100 ' 10* 10*10* 10' 10 (d) ' 10' 10*10* 10' 10 * (e) Figure * 10 * 10* 10* 10' 10* Average Nusselt Numbers (a) UH= 1, for Geometry (a); (b) UH=1, for Geometry (b); (c) L/H=1, for Geometry (c); (d) UH=2, for Geometry (a); (e) L/H=2, for Geometry (b); (f) L/H=2, for Geometry (c);

13 207 ' i ' 10 * 10 * 10 * 10' ' 10 * 10 * 10 * 10 ' 10 ' > y i ' * 10*10' 10' 10" (c) * 10* 10*10* 10' 10 (d) 100i 10 y r i ' 10* 10*10* 10' 10 (e) ' 10* 10*10* 10' 10 (f) Figure 5. Average Exchange Flow te (a) L/H=1, for Geometry (a); (b) L/H=1, for Geometry (b); (c) L/H=1, for Geometry (c); (d) L/H=2, for Geometry (a); (e) L/H=2, for Geometry (b); (f) L/H=2, for Geometry (c);

14 208 Air Pollution ' h/h Figure 6. The Effects of Aperture tios(h/h) on Exchange Flow te for Geometry (b); for L/H=1; for L/H=2.