A Numerical Study of the Convective Heat Transfer Rate from a Window Covered by a Slatted Top Down- Bottom Up Blind System to an Adjacent Room

P.H. Oosthuizen and J. T. Paul

Department of Mechanical and Materials Engineering, Queen’s University Kingston, ON, Canada K7L 3N6

Email: oosthuiz@me.queensu.ca

ABSTRACT

Top Down–Bottom Up blinds have become popular in recent times. However, the effects of such blind systems on the convective heat transfer from the window to the surrounding room have not been extensively studied. The purpose of the present work, therefore, is to numerically investigate the effect of Top Down–Bottom Up slatted blinds on this convective heat transfer. An approximate model of the window-blind system has been adopted. The “window” considered is recessed and is assumed to be at a uniform surface temperature and to be exposed to a room in which the air temperature far from the window is uniform. The blind slats are assumed to be thin and to offer no resistance to heat transfer. Fluid properties have been assumed constant except for the density change with temperature that gives rise to the buoyancy forces this being treated by using the Boussinesq approach. Radiant heat transfer effects have been neglected. The governing equations have been solved using the commercial finite-volume based CFD code FLUENT

. The standard k-epsilon turbulence model has been used with full account being taken of the effect of the buoyancy forces. The main emphasis in this study is on the effect of the size of the blind openings at the top and bottom of the window and on the effect of the gap between the blind and the window on the convective heat transfer rate from the window to the room.

1. INTRODUCTION

Improved models for the convective heat transfer rate from the inner surface of a window to the surrounding room for the case where the window is fully or partially covered by a blind are needed to assist in the development of systems that reduce the overall heat transfer rate through the window. Top Down–Bottom Up blinds have become quite popular

in recent times and have the potential to reduce energy consumption by allowing the controlled use of sunlight to illuminate the house (daylighting) and/or to use passive solar room heating while still providing shade and privacy to the occupants. However, the effects of such blind systems on the convective heat transfer from the window to the surrounding room have not been extensively studied. The purpose of the present work was, therefore, to numerically investigate the effect of the size of the top and bottom blind openings with Top Down–Bottom Up slatted blinds on the convective heat transfer rate from the window and to investigate the effect of the blind-to-window gap on this convective heat transfer rate. A recessed window with a Top Down–Bottom Up blind of the type here being considered is shown in Fig. 1. With the type of blind

Figure 1 Flow situation considered

here being considered the blind opening is achieved by adjusting the angle that the interconnected blind slats make to the vertical, the greater the total blind opening the closer the slats are to the horizontal and therefore the closer they are to the window. Top Down–Bottom Up blinds, in general, will have different blind openings at the top and at the bottom as shown in Fig. 1. The sizes of both the top and bottom openings will affect the convective heat transfer rate from the window to the room and this has been numerically investigated in the present study. The heat transfer rate from the window will also be influenced by the recess depth of the window, d (see Fig. 2). When the top and bottom blind openings are changed the gap between the blind and the window for a given dimensionless recess depth changes producing changes in the heat transfer from the window.

There have been many studies of the effect of blinds on the heat transfer rate between the room-side of the window and the room but these have mainly been concerned with traditional Bottom-Up type blinds. Typical of such studies of the case where plane blinds are being used are those of Oosthuizen et al. [1, 2] and Oosthuizen [3, 4]. These studies are for the case where the flow over the window-blind system is laminar. The effect of flow transition for the case of flow over a plane blind has been considered by, for example, Oosthuizen and Naylor [5], and Oosthuizen [6, 7]. Typical of the studies of the case where Venetian blinds are used are those of Collins et al. [8, 9], Duarte et al. [10], Machin et al. [11], Shahid [12], and Roeleveld et al. [13]. Existing studies of Top Down–Bottom Up blinds have concentrated on plane blinds, e.g., see Oosthuizen and Paul [14]. The present study, and also many of the previous studies, related to the effects of blinds on window heat transfer, considers only the convective heat transfer. In window heat transfer situations, the radiant heat transfer can be very important and can interact with the convective flow, e.g. see Phillips et al. [15].

2. SOLUTION PROCEDURE

The mean flow has been assumed to be steady and the Boussinesq approach has been used. It has been assumed that the “window” is at a uniform temperature, TW, and that this window temperature is higher than the temperature, TF , of the air in the “room” to which the window is exposed. The solution has been obtained by numerically solving the governing equations subject to the boundary conditions using the commercial CFD solver, FLUENT. The flow has been assumed to be two-dimensional. In the situation considered here both laminar and turbulent flow can occur. The k-ε turbulence model with the full effects of buoyancy forces accounted for and with standard wall functions has been used in obtaining the solutions. This turbulence model has been found in past studies to give moderately good predictions of when transition to turbulence occurs and of the flow and heat transfer in the laminar, transitional, and turbulent regions. Extensive grid - and convergence criterion independence testing was undertaken and indicated that the heat transfer results presented here are to within 1% independent of the number of grid points and of the convergence-criterion used.

The mean convective heat transfer rates from the window have been expressed in terms of the mean Nusselt number defined by:

(1)

where q is the mean heat transfer rate from the

window considered, k is the thermal conductivity, TW

is the window surface temperature, TF is the room air temperature, and L is the height of the window. The Rayleigh number used is in presenting the results is also based on L and on the overall temperature difference between the window temperature and the room air temperature, i.e., is defined by:

(2)

where β, , and are the bulk expansion coefficient, the density, and the viscosity respectively.

3. RESULTS

The solution parameters are:

1. the Rayleigh number, Ra ,

2. the Prandtl number, Pr ,

3. the dimensionless “depth” to which the window is recessed, D = d / L, r being the window recess depth,

Figure 2 Definitions of blind slat size

and recess depth

/ ( )W FNu q L k T T

2 3 /p W FRa g c T T L k

4. the dimensionless top and bottom blind openings, Ht = ht / L and Hb = hb / L, where L is the overall height of the window and ht and hb are the top and blind openings as shown in Fig. 1,

5. the number of slats, n, the slat size, s , being taken as L / n where n is the number of slats. The top and bottom slats have, as indicated in Fig.1, size s/2, s being the size of the full slats, n is the number of equivalent full slats.

Consideration is first given to the case where the top and bottom blind openings are the same, i.e., where Ht = Hb. Typical results for this case are shown in Figs. 3, 4, and 5 which show variations of Nusselt number with Rayleigh number for dimensionless top and bottom openings of 0.1, 0.2, and 0.3. It will be seen from these figures that at the higher Rayleigh numbers in the laminar flow region the mean Nusselt

number decreases with decreasing dimensionless recess depth due to the interference of the slats with the flow over the “window”. However, at the higher Rayleigh numbers in the turbulent flow region the dimensionless recess depth has essentially has no effect on the mean Nusselt number because of the thin boundary layer that exists under this situation. At the lower Rayleigh numbers in the laminar flow region the mean Nusselt number will be seen to increase with decreasing dimensionless recess depth due to the thick boundary layer that then exists and the fact that the flow is accelerated between the slats and the “window”.

To illustrate the effect of having different top and bottom blind openings on the form of the variation of mean Nusselt number with Rayleigh number results are given in Fig. 6 for the case where the dimensionless top and bottom openings are 0.3 and

Figure 5 Variation of Mean Nusselt number with Rayleigh number for dimensionless top and bottom

blind openings of 0.3 for two values of the dimensionless recess depth

Figure 6 Variation of Mean Nusselt number with Rayleigh number for dimensionless top and bottom blind openings of 0.3 and 0.1 respectively for two

values of the dimensionless recess depth

Figure 3 Variation of Mean Nusselt number with Rayleigh number for dimensionless top and bottom

blind openings of 0.1 for three values of the dimensionless recess depth

Figure 4 Variation of Mean Nusselt number with Rayleigh number for dimensionless top and bottom

blind openings of 0.2 for two values of the dimensionless recess depth.

0.1 respectively. It will be seen from these results that the basic form of the Nusselt number–Rayleigh number variation is the same as that which exists when the top and bottom blind openings are the same.

Attention will next be turned to the local heat transfer rate variation up the window. The heat transfer rate has been expressed in terms of the local Nusselt number based on the window height, h, i.e., in terms of:

(3)

where q’wl is the local heat transfer rate per unit area.

Typical variations of the local Nusselt number with dimensionless distance up the window, Y, for dimensionless top and bottom blind openings of 0.1, a dimensionless window recess depth of 0.05, and Rayleigh numbers of 1011, 109, and 107 are shown in Figs. 7, 8, and 9 respectively. The peaks in the local Nusselt number variation occur at the points where the blind is closest to the window. The peaks will be seen to grow sharper as the Nusselt number decreases due to the increase in the thickness of the boundary layer on the window as the Rayleigh number decreases.

The minimum gap between the window and the blind decreases as the total dimensionless blind opening (Wtop + Wbot) increases. The effect of this can be seen by comparing the results given in Figs. 10 and 11

Figure 9 Variation of local Nusselt number based on window height, h, with dimensionless distance up the window Y for dimensionless top and bottom openings of 0.1, a dimensionless window recess depth of 0.05,

and a Rayleigh number of 107

Figure 10 Variation of local Nusselt number based on window height, h, with dimensionless distance up the window Y for dimensionless top and bottom openings of 0.3, a dimensionless window recess depth of 0.05,

and a Rayleigh number of 109

Figure 7 Variation of local Nusselt number based on window height, h, with dimensionless distance up the

window Y for dimensionless top and bottom openings of 0.1, a dimensionless window recess depth of 0.05, and a Rayleigh number of 1011.

Figure 8 Variation of local Nusselt number based on window height, h, with dimensionless distance up the window Y for dimensionless top and bottom openings of 0.1, a dimensionless window recess depth of 0.05,

and a Rayleigh number of 109

( )wl

lw f

Nuq h

T T k

with those given in Figs 8 and 9. The results given in Figs. 10 and 11 are for dimensionless top and bottom blind openings of 0.3, a dimensionless window recess depth of 0.05, and Rayleigh numbers of 109 and 107 respectively. It will be seen that as the total window opening increases the sharpness of the peaks in the local Nusselt number distribution increases.

The effect of having different top and bottom blind openings is illustrated by the results given in Figs. 12 and 13. These figures give results for dimensionless top and bottom blind openings of 0.3 and 0.1 respectively, a dimensionless window recess depth of 0.05, and Rayleigh numbers of 1011 and 107 respectively.

The results given in Figs. 7 to 13 were all for a dimensionless window recess depth of 0.05. The effect of the window recess depth on the local heat transfer distributions can be seen by comparing the results given in Figs. 14, 15, and 16 which are for a dimensionless window recess depth of 0.07, with those given in Figs. 7 to 13. The results in Figs 14, 15, and 16 are for dimensionless top and bottom blind openings of 0.3 and 0.1 respectively, and Rayleigh numbers of 1011, 109 and 107 respectively. It will be seen that with the larger dimensionless window recess depth the sharpness of the peaks in the local Nusselt number distribution is reduces and that at the largest Rayleigh number considered the changes in the local Nusselt number with Y are relatively small.

CONCLUSIONS

The results of the present study indicate that:

1. When the dimensionless top and bottom blind openings are equal the mean heat transfer rate from the window increases with increasing dimensionless blind opening at the lower Rayleigh numbers considered and decreases with increasing dimensionless blind opening at the higher Rayleigh numbers considered

2. When the dimensionless top and bottom blind openings are different the mean heat transfer rate from the window is mainly dependent on the total dimensionless blind opening.

3. The main influence of the dimensionless window recess depth on the mean heat transfer rate from the window is at the intermediate Rayleigh numbers considered, i.e., in the transition region.

4. When the blind is open at the top and bottom there are large variations in the local heat transfer rate variation with distance up the window, the highest heat transfer rates occurring at the points where the blind is closest to the window.

ACKNOWLEDGEMENTS

This work was supported by the Natural Science and Engineering Research Council (NSERC) of Canada.

Figure 12 .Variation of local Nusselt number based on window height, h, with dimensionless distance

up the window Y for dimensionless top and bottom openings of 0.3 and 0.1 respectively,

a dimensionless window recess depth of 0.05, and a Rayleigh number of 1011

Figure 11 Variation of local Nusselt number based on window height, h, with dimensionless distance

up the window Y for dimensionless top and bottom openings of 0.3, a dimensionless

window recess depth of 0.05, and a Rayleigh number of 107

REFERENCES

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Figure 13 Variation of local Nusselt number based on window height, h, with dimensionless distance

up the window Y for dimensionless top and bottom openings of 0.3 and 0.1 respectively,

a dimensionless window recess depth of 0.05, and a Rayleigh number of 107

Figure 14 .Variation of local Nusselt number based on window height, h, with dimensionless distance

up the window Y for dimensionless top and bottom openings of 0.3 and 0.1 respectively,

a dimensionless window recess depth of 0.07, and a Rayleigh number of 1011

Figure 15 Variation of local Nusselt number based on window height, h, with dimensionless

distance up the window Y for dimensionless top and bottom openings of 0.3 and 0.1 respectively, a dimensionless window recess depth of 0.07,

and a Rayleigh number of 109

Figure 16 Variation of local Nusselt number based on window height, h, with dimensionless distance up the window Y for dimensionless top and bottom openings of 0.3 and 0.1 respectively, a dimensionless window recess depth of 0.07, and a Rayleigh number of 107

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