Draft:Natural Convection in Partially Divided Enclosures

Natural Convection in Partially Divided Enclosures

Introduction

In real-world systems such as buildings, lakes, and solar collectors, natural convection often does not conform to the simplified single-enclosure models found in much of the literature. Instead, these systems can be represented as partially divided enclosures. A basic model for studying natural convection in these systems involves two enclosures that communicate laterally through openings such as doorways, windows, or corridors, or through incomplete dividing walls. This model has been the subject of various studies and experiments that examine the two-dimensional geometries of these enclosures.

Flow Features in Partially Divided Enclosures

The introduction of a vertical obstacle within the cavity creates a distinct flow feature: the trapping of fluid on one side of the obstacle. This fluid becomes largely stagnant and inactive in terms of convective heat transport, leading to significant changes in the overall flow patterns. For example, when a partial wall is positioned on the floor of the cavity, the fluid on the cold side becomes trapped, and its circulation is severely restricted. This contrasts with a scenario where no internal obstructions exist, in which the flow can circulate freely throughout the entire cavity.

The presence of these significant pools of inactive or stagnant fluid, as shown in Figure 1, leads to a noticeable reduction in the heat transfer rate across the enclosure. This occurs because the stagnant fluid acts as a thermal barrier, reducing the efficiency of heat exchange between the cold and hot sides of the cavity. The overall thermal performance of the enclosure is thus significantly impacted by the size and placement of the internal partitions, which alter the fluid’s movement and convective heat transport. Understanding these flow features is essential for optimizing systems like HVAC enclosures or solar collectors, where thermal management is crucial.

Heat Transfer Reduction

The reduction in end-to-end heat transfer can be predicted through scale analysis. Assuming no heat transfer occurs through the incomplete partition, the thermal resistance from the two boundary layers near the heated walls impedes heat transfer. Let $(\delta _{1},H_{1})$ and $(\delta _{2},H_{2})$ represent the length scales of the two boundary layers, as shown in Figure 1. The heights $H_{1}$ and $H_{2}$ are determined by the geometry of the internal partition.

For fluids where the Prandtl number $(Pr>1)$, the thermal boundary layer thicknesses are approximated as:

$\delta _{1}\sim H_{1}Ra_{H1}^{-1/4},\quad \delta _{2}\sim H_{2}Ra_{H2}^{-1/4}$

Here, the subscripts $H_{1}$ and $H_{2}$ indicate that the Rayleigh number is based on the actual height of the boundary layer, not the overall height of the enclosure $H$. The thermal resistances for the two boundary layers can be expressed as:

${\frac {\delta _{1}}{kH_{1}}}\quad {\text{and}}\quad {\frac {\delta _{2}}{kH_{2}}}$

where $k$ is the thermal conductivity of the fluid.

End-to-End Heat Transfer Calculation

The total heat transfer rate $q'$ can be computed as follows:

$q={\frac {\Delta T}{\left({\frac {C_{1}\delta _{1}}{kH_{1}}}+{\frac {C_{2}\delta _{2}}{kH_{2}}}\right)}}$

where $C_{1}$ and $C_{2}$ are numerical coefficients of order $O(1)$. This equation can be rearranged into dimensionless form to illustrate the influence of the obstacle geometry:

${\frac {q'}{k\Delta T}}={\frac {Ra_{H}^{1/4}}{C_{1}\left({\frac {H}{H_{1}}}\right)^{3/4}+C_{2}\left({\frac {H}{H_{2}}}\right)^{3/4}}}$

Looking at Figure 1b and above equation, we see that as the opening left above the partition decreases, the ratio ${\frac {H}{H_{2}}}$ becomes significantly larger, resulting in a sharp decline in the heat transfer rate. For instance, setting $C_{1}=1.5$ and $C_{2}=3$, the equation aligns well with heat transfer rates measured in a box with a single internal partition (as in Fig. 1b, where $H=H_{1}$), particularly in the range of Rayleigh numbers $10^{9}$ to $10^{10}$.

Fluid Trap Phenomenon

The fluid trap phenomenon is a critical factor in thermal dynamics, occurring when internal partitions within an enclosure inhibit natural circulation. This effect significantly influences heat transfer and overall system performance. When a floor obstacle is placed on the hot side of the enclosure, as illustrated in Figure 2a, the fluid trapped between two obstacles becomes stably stratified, preventing convection. This stagnant layer results in poor thermal exchange, as the warmer fluid cannot mix with the cooler fluid above it.

Conversely, when the obstacle is positioned on the cold side (Figure 2b), natural convection can still occur, but the flow may be constrained to a single cell. This restricted movement illustrates how the configuration of internal structures dictates fluid behavior and thermal efficiency. The implications of fluid trapping are particularly relevant in applications such as building design and solar collectors, where maximizing heat transfer is crucial.

Research has demonstrated that strategic placement of obstacles can optimize thermal performance. For example, studies by Zhang et al. on venetian blinds as vertical permeable screens show how adjustable barriers can control fluid movement, enhancing energy efficiency. Understanding and leveraging the fluid trap phenomenon is essential for designing effective thermal systems in various practical applications.

Conclusion

Partially divided enclosures provide a complex yet insightful model for studying natural convection in real-world systems. The presence of partitions not only introduces new flow features, such as fluid trapping, but also significantly impacts the overall heat transfer rates. By understanding the thermal resistances and the governing equations, we can better predict and optimize heat transfer in various applications, ranging from building design to solar collectors.

References

• Bejan, A. (2013). Convection Heat Transfer (4th ed.). Wiley.
• Zhang, Z., Bejan, A., & Lage, J. L. (1991). Natural convection in a vertical enclosure with internal permeable screen. Journal of Heat Transfer, Vol. 113, pp. 377–383.
• Incropera, F. P.; DeWitt, D. P.; Bergman, T. L.; Lavine, A. S. (2007). Fundamentals of Heat and Mass Transfer (6th ed.). Wiley.
• Kays, W. M.; Crawford, M. E. (1993). Convective Heat and Mass Transfer (3rd ed.). McGraw-Hill.