# Rectangular Enclosures

Heat transfer in regimes I and II, shown in Table 1 from Scale Analysis2, is dominated by conduction and the fluid circulation plays an insignificant role. Therefore, heat transfer in these two regimes is practically equal to heat transfer rate estimated using Fourier’s law.

Figure 1: Natural convection in a square enclosure (RaH = 106, Pr=0.71). Steady-state distributions of dimensionless (a) temperature, (b) pressure, (c) horizontal velocity, and (d) vertical velocity

The flow pattern in regime III is of the boundary layer type and the core of the fluid is stagnant and stratified. The fluid flow and heat transfer in regime III can be obtained via boundary layer analysis. Figure 1 shows contours of dimensionless temperature, pressure, horizontal, and vertical velocities for natural convection in a square enclosure (H / L = 1) heated from its left side and cooled from its right side while insulated from the top and bottom. The behavior of boundary layer type flow and heat transfer is apparent because both temperature and velocity gradient peak near the heated and cooled wall. The low gradients of temperature and velocity in the middle of the enclosure further indicate a stagnant and stratified core. For high Rayleigh number, the average Nusselt number for regime III, as obtained from boundary layer analysis, is (Bejan, 2004):

${{\overline{\text{Nu}}}_{H}}=0.364\frac{L}{H}\text{Ra}_{H}^{1/4} \qquad \qquad(1)$

which indicates that natural convection heat transfer in a rectangular enclosure is dominated by both the Rayleigh number and aspect ratio.

For laminar natural convection in a rectangular enclosure with aspect ratio H/L between 1 and 10, the following correlations were suggested by Berkovsky and Polevikov (1977):

${{\overline{\text{Nu}}}_{H}}=0.22{{\left( \frac{\Pr }{0.2+\Pr }\text{R}{{\text{a}}_{H}} \right)}^{0.28}}{{\left( \frac{L}{H} \right)}^{0.09}} \qquad \qquad(2)$

which is valid for $2, and

${{\overline{\text{Nu}}}_{H}}=0.18{{\left( \frac{\Pr }{0.2+\Pr }\text{R}{{\text{a}}_{H}} \right)}^{0.29}}{{\left( \frac{L}{H} \right)}^{-0.13}} \qquad \qquad(3)$

which is valid for 1 < H / L < 2,10 − 3 < Pr < 103 < [Pr / (0.2 + Pr)]RaH(L / H)3. Catton (1978) found that the above correlations agreed well with the experimental data and the results from numerical solutions.

For a rectangular enclosure with a larger aspect ratio, MacGregor and Emery (1969) recommended the following correlations:

${{\overline{\text{Nu}}}_{L}}=0.42\text{Ra}_{L}^{1/4}{{\Pr }^{0.012}}{{\left( \frac{H}{L} \right)}^{-0.3}} \qquad \qquad(4)$

which is valid for $10 < H/L < 40,{\rm{ 1 < }}\Pr < 2 \times {10^4},{\rm{ 1}}{{\rm{0}}^{\rm{4}}}{\rm{ < R}}{{\rm{a}}_L} < {10^7}$, and

${{\overline{\text{Nu}}}_{L}}=0.046\text{Ra}_{L}^{1/3} \qquad \qquad(5)$

which is valid for $1 < H/L < 40,{\rm{ 1 < }}\Pr < 20,{\rm{ }}{10^6} < {\rm{R}}{{\rm{a}}_L} < {10^9}$. It should be pointed out that the characteristic length for the Rayleigh and Nusselt numbers used in eqs. (4) and (5) is the width of the enclosure, L.

For the shallow enclosure represented by regime IV in Table 1 from Scale Analysis2, the horizontal wall jet flows from the left to right on the top wall and from right to the left on the bottom wall. Figure 2 shows streamlines and isotherms for natural convection in a shallow enclosure (H / L = 0.1) heated from the right side

Figure 2; Natural convection in a shallow enclosure: (a) RaH = 106,(b)RaH = 108 (H/L = 0.1, Pr = 1.0).

and cooled from the left side (Tichy and Gadgil, 1982). It can be observed from these figures that there exist wall jets on both the top and bottom walls and that the high velocity in these wall jets drives the core flow. The entire enclosure can be divided into two end regions near the heated and cooled vertical walls and a core region between the two end regions. There are large temperature gradients along the horizontal direction in the end regions, but the core region is essentially stratified. Under fixed aspect ratio, increasing Rayleigh number makes the horizontal wall jets thinner. The core region extends almost the entire length of the shallow enclosure for high Rayleigh number.

Bejan and Tien (1978) performed a scale analysis and obtained an analytical solution via analysis of fluid flow and heat transfer in the core region. As indicated by Fig. 3, their results compared favorably with experimental (Imberger, 1973) and numerical (Cormack et al., 1974) results. Bejan (2004) pointed out that Fig. 3 is not only useful for regime IV of Table 1 from Scale Analysis2 (shallow enclosure), but also for regime III (boundary layer regime). The success of Bejan and Tien’s analysis was attributed to the fact that the thermal resistances associated with the vertical end walls were taken into account. For a limited case, the asymptotic heat transfer results can be expressed as:

${{\overline{\text{Nu}}}_{H}}=1+\frac{1}{362880}{{\left( \frac{H}{L}\text{R}{{\text{a}}_{H}} \right)}^{2}} \qquad \qquad()$

which is valid when ${{(H/L)}^{2}}\text{R}{{\text{a}}_{H}}\to 0$, and is shown as the dashed line in Fig. 3.

Figure 3: Average Nusselt number for natural convection in a shallow enclosure

## Contents

#### Heated from the Bottom

For a rectangular enclosure filled with fluid and heated from the side, natural convection will be initiated as soon as the temperature difference between the two vertical walls is established. For a rectangular enclosure heated from below, natural convection may or may not occur depending on whether the temperature difference between the top and bottom walls exceeds a critical value. If the temperature difference is not sufficient to initiate natural convection, the mechanism of heat transfer is by conduction of the fluid (and radiation for the case where the fluid is a gas). The condition for the onset of natural convection can be expressed in terms of a critical Rayleigh number. For the case that the horizontal dimension is much larger than the height of the enclosure, the criterion for the onset of natural convection is:

$R{{a}_{H}}=\frac{g\beta \Delta T{{H}^{3}}}{\nu \alpha }>1708 \qquad \qquad(7)$

When the Rayleigh number just exceeds the above critical Rayleigh number, the flow pattern is two-dimensional counter rotating rolls – referred to as Bénard cells [see Fig. 4 (a)]. As the Rayleigh number further increases to one or two orders of magnitude higher than the above critical Rayleigh number, the two-dimensional cells breakup to three dimensional cells whose top view is hexagonal [see Fig. 4(b)]. The function of the two-dimensional rolls and three-dimensional hexagonal cells is to promote heat transfer from the heated bottom wall to the cooled top wall. Globe and Dropkin (1959) suggested the following empirical correlation

${{\overline{\text{Nu}}}_{H}}=0.069\text{Ra}_{H}^{1/3}{{\Pr }^{0.074}} \qquad \qquad(8)$

Figure 4: Rolls and hexagonal cells in natural convection in enclosure heated from below

where all thermophysical properties are evaluated at (Th + Tc) / 2. Equation (8) is valid for $3\times {{10}^{5}}<\text{R}{{\text{a}}_{H}}<7\times {{10}^{9}}$. In addition, H/L must be sufficiently small so that the effect of the sidewalls can be negligible.

#### Inclined Rectangular Enclosure

When the rectangular enclosure heated from the side is tilted relative to the direction of gravity, additional unstable stratification and thermal instability, which are usually associated with natural convection in an enclosure heated from below, will affect the fluid flow and heat transfer. The tilt angle γ [see Fig. 1 from Natural Convection in Enclosures (c)] becomes an additional parameter that plays a significant role. When γ is increased from 0° to 90°, the heat transfer mechanism changes from pure conduction at $\gamma ={{0}^{\circ }}$ (heated from the top) to single-cell convection at $\gamma ={{90}^{\circ }}$ (heated from the side), where both buoyancy effect and Nusselt number are maximized. As the inclination angle γ is further increased beyond 90°, the Nusselt number starts to decrease until a local minimum at γ = γc is reached. The value of γc depends on the aspect ratio of the enclosure (see Table 1). Beyond γc, the stable two-dimensional single-cell convection becomes unstable and transits to three-dimensional rolls. The mechanism of Bénard convection sets in, which results in an increase of the Nusselt number. The variation of Nusselt number as a function of tilt angle is qualitatively shown in Fig. 5.

Table 1: Critical inclination angle for different aspect ratio (Arnold et al., 1976)
 Aspect ratio, ''H / L'' 1 3 6 12 >12 Critical tilt angle, δc 155° 127° 120° 113° 110°
Figure 5: Effect of inclination angle on natural convection in enclosure

Figure 6: Natural convection in inclined squared enclosures

Zhong et al. (1983) numerically solved the natural convection in an inclined square enclosure, and the isotherms and the streamlines for Ra = 105 are shown in Fig. 6. The transition from a conduction dominated regime at $\gamma ={{45}^{\circ }}$ to single-cell convection at $\gamma ={{90}^{\circ }}$ is apparent from the isotherms. At $\gamma ={{135}^{\circ }}$, which, according to Table 1, is less than the critical inclination angle, the isotherms start to exhibit some behaviors of thermally unstable conditions. They proposed the following correlation for natural convection of air in a square enclosure (H / L = 1) in the region $0<\gamma <{{90}^{\circ }}$:

${{K}_{\gamma }}=\frac{{{\overline{\text{Nu}}}_{H}}(\gamma )-{{\overline{\text{Nu}}}_{H}}({{0}^{\circ }})}{{{\overline{\text{Nu}}}_{H}}({{90}^{\circ }})-{{\overline{\text{Nu}}}_{H}}({{0}^{\circ }})}=\frac{2}{\pi }\gamma \sin \gamma \qquad \qquad(9)$

where

${{\overline{\text{Nu}}}_{H}}({{0}^{\circ }})$

is for pure conduction. While eq. (9) is good for air in a squared enclosure, the following correlation can be applied to other situations:

$\frac{L}{H}{{\overline{\text{Nu}}}_{H}}(\gamma )=\left\{ \begin{matrix} 1+\left[ \frac{L}{H}{{\overline{\text{Nu}}}_{H}}({{90}^{\circ }})-1 \right]\sin \gamma & 0<\gamma <{{90}^{\circ }} \\ \frac{L}{H}{{\overline{\text{Nu}}}_{H}}({{90}^{\circ }}){{(\sin \gamma )}^{1/4}}\text{ } & {{90}^{\circ }}<\gamma <{{\gamma }_{c}} \\ \end{matrix} \right. \qquad \qquad(10)$

which indicates that the Nusselt number for an inclined enclosure is related to the Nusselt number at $\gamma ={{90}^{\circ }}$ and the actual inclination angle.

$q({{0}^{\text{o}}})=kDH\frac{{{T}_{h}}-{{T}_{c}}}{L}=0.0263\times 0.5\times 0.5\times \frac{37-17}{0.05}=2.63\text{W}$

Figure 7: Natural convection in inclined squared enclosure.

The Nusselt number for the case of pure conduction is ${{\overline{\text{Nu}}}_{L}}({{0}^{\text{o}}})=1$or ${{\overline{\text{Nu}}}_{H}}({{0}^{\text{o}}})=(H/L){{\overline{\text{Nu}}}_{L}}({{0}^{\text{o}}})=10$. Heat transfer for the case of γ = 45o can be calculated using eq. (10), which requires the Nusselt number for γ = 90o. Thus, the heat transfer rate for γ = 90o will be calculated first. The Rayleigh number is:

$\text{R}{{\text{a}}_{H}}=\frac{g\beta ({{T}_{h}}-{{T}_{c}}){{H}^{3}}}{\nu \alpha }=\frac{9.807\times 1/300\times (37-17)\times {{0.5}^{3}}}{15.89\times {{10}^{-6}}\times 22.5\times {{10}^{-6}}}=2.286\times {{10}^{8}}$

The Nusselt number for γ = 90o can be obtained from eq. (3):

\begin{align} & {{\overline{\text{Nu}}}_{H}}({{90}^{\text{o}}})=0.18{{\left( \frac{\Pr }{0.2+\Pr }\text{R}{{\text{a}}_{H}} \right)}^{0.28}}{{\left( \frac{L}{H} \right)}^{-0.13}} \\ & =0.18\times {{\left( \frac{0.707}{0.2+0.707}\times 2.286\times {{10}^{8}} \right)}^{0.28}}{{\left( \frac{0.05}{0.5} \right)}^{-0.13}}=49.6 \\ \end{align}

The heat transfer coefficient is:

$h({{90}^{\text{o}}})=\frac{k{{\overline{\text{Nu}}}_{H}}({{90}^{\text{o}}})}{H}=\frac{0.0263\times 49.6}{0.5}=2.61\text{ W/}{{\text{m}}^{\text{2}}}\text{-K}$

Therefore, the heat transfer rate for γ = 90o is:

$q({{90}^{\text{o}}})=h({{90}^{\text{o}}})DH({{T}_{h}}-{{T}_{c}})=2.61\times 0.5\times 0.5\times (37-17)=6.53\text{ W}$

When the inclination angle is γ = 45o = π / 4, eq. (10) yields:

\begin{align} & \frac{L}{H}{{\overline{\text{Nu}}}_{H}}({{45}^{\text{o}}})=1+\left[ \frac{L}{H}{{\overline{\text{Nu}}}_{H}}({{90}^{\circ }})-1 \right]\sin \gamma \\ & =1+\left( \frac{0.05}{0.5}\times 49.6-1 \right)\sin {{45}^{\text{o}}}=3.80 \\ \end{align}

Thus, the Nusselt number is

NuH(45o) = 38.0

and the corresponding heat transfer coefficient is:

$h({{45}^{\text{o}}})=\frac{k{{\overline{\text{Nu}}}_{H}}({{45}^{\text{o}}})}{H}=\frac{0.0263\times 38.0}{0.5}=2.00\text{ W/}{{\text{m}}^{\text{2}}}\text{-K}$

The heat transfer rate for γ = 45o is therefore:

$q({{45}^{\text{o}}})=h({{45}^{\text{o}}})DH({{T}_{h}}-{{T}_{c}})=2\times 0.5\times 0.5\times (37-17)=5.00\text{W}$

When the inclination angle is γ = 180o, the problem becomes natural convection in an enclosure heated from the below. The Rayleigh number is:

$\text{R}{{\text{a}}_{L}}=\frac{g\beta ({{T}_{h}}-{{T}_{c}}){{L}^{3}}}{\nu \alpha }=\frac{9.807\times 1/300\times (37-17)\times {{0.05}^{3}}}{15.89\times {{10}^{-6}}\times 22.5\times {{10}^{-6}}}=2.286\times {{10}^{5}}$

The Nusselt number in this case can be obtained from eq. (8):

\begin{align} & {{\overline{\text{Nu}}}_{L}}({{180}^{\text{o}}})=0.069\text{Ra}_{H}^{1/3}{{\Pr }^{0.074}} \\ & =0.069\times {{(2.286\times {{10}^{5}})}^{1/3}}\times {{0.707}^{0.074}}=4.11 \\ \end{align}

The heat transfer coefficient is

$h({{180}^{\text{o}}})=\frac{k{{\overline{\text{Nu}}}_{L}}({{180}^{\text{o}}})}{L}=\frac{0.0263\times 4.11}{0.05}=2.16\text{ W/}{{\text{m}}^{\text{2}}}\text{-K}$

The heat transfer rate for γ = 180o is therefore,

$q({{180}^{\text{o}}})=h({{180}^{\text{o}}})DH({{T}_{h}}-{{T}_{c}})=2.16\times 0.5\times 0.5\times (37-17)=10.80\text{ W}$

## References

Arnold, J.N., Catton, I., and Edwards, D.K., 1976, “Experimental Investigation of Natural Convection in Inclined Rectangular Regions of Differing Aspect Ratios,” ASME J. Heat Transfer, Vol. 98, pp. 67-71.

Bejan, A., 2004, Convection Heat Transfer, 3rd ed., John Wiley & Sons, New York.

Bejan, A., and Tien, C.L., 1978, “Laminar Natural Convection Heat Transfer in a Horizontal Cavity with Different End Temperatures,” ASME J. Heat Transfer, Vol. 100, pp. 641-¬647.

Berkovsky, B.M. and Polevikov, V.K., 1977, “Numerical Study of Problems on High-Intensive Free Convection,” Heat Transfer and Turbulent Buoyant Convection, Eds. Spalding, D.B., and Afgan, N., Vol. II, pp. 443-455, Hemisphere Publishing Corporation, Washington, DC.

Caton, I., 1978, “Natural Convection in Enclosures,” Proceedings of the 6th International Heat Transfer Conference, Vol. 6, pp. 13-43, Toronto, Canada.

Cormack, D. E., Leal, L. G., and Seinfeld, J. H., 1974, “Natural Convection in a Shallow Cavity with Differentially Heated End Walls: Part 2. Numerical Solutions,” J. Fluid Mech., Vol. 65, pp. 231-246.

Globe, S., and Dropkin, D., 1959, “Natural Convection Heat Transfer in Liquids Confined by Two Horizontal Plates Heated from Below,” ASME J. Heat Transfer, Vol. 81, pp. 24-28.

Imberger, I., 1973, “Natural Convection in a Shallow Cavity with Differentially Heated End Walls, Part 3. Experimental Results,” J. Fluid Mech., Vol. 65, pp. 247-260.

MacGregor, R.K., and Emery, A.P., 1969, “Free Convection through Vertical Plane Layers - Moderate and High Prandtl Number Fluids,” ASME J. Heat Transfer, Vol. 91, pp. 391-403

Tichy, J., and Gadgil, A., 1982, “High Rayleigh Number Laminar Convection in Low Aspect Ratio Enclosures with Adiabatic Horizontal Walls and Differentially Heated Vertical Walls,” ASME J. Heat Transfer, Vol. 104, pp. 103-110.

Zhong, Z.Y., Lloyd, J.R., Yang, K.T., 1983, “Variable-Property Natural Convection in Tilted Square Cavities,” Numerical Method in Thermal Problems, Vol. III, Eds. Lewis, R.W., Johnson J.A, and Smith W.R., pp. 968-979, Pineridge Press, Swansea, UK.