Free Convection in Enclosures
Calculate heat transfer inside cavities, rectangular slots, double-pane windows, and concentric cylinders or spheres.
Cavity Natural Convection
In enclosures, fluid movement is confined. For low Rayleigh numbers ($\text{Ra} \le 10^3$), heat transfer is purely conductive ($\text{Nu} \approx 1.0$). At higher Rayleigh numbers, convective circulation cells form, which significantly increases heat exchange.
Parameters Setup
Results & Curves
📈 Nusselt number vs Delta T (dT)
Engine Output
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FREE CONVECTION IN ENCLOSURES
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--- INPUTS ---
Enclosure Type = 4
Gap L = 0.0000 m
Height H = 0.0000 m
Width W = 0.0000 m
T_hot = 100.00 C
T_cold = 30.00 C
Delta T = 70.00 C
--- FLUID ---
rho = 1.1770 kg/m3
mu = 1.8500E-05 Pa.s
k = 0.026300 W/mK
Pr = 0.7100
beta = 2.9573E-03 1/K
--- RESULTS ---
Rayleigh Number Ra = 9.0946E+04
Nusselt Number Nu = 5.4961
Effective h = 5.7819 W/m2K
k_effective = 0.1445 W/mK
Heat Transfer Q = 91.7196 W (or W/m)
--- DELTA-T SWEEP ---
dT[C] Ra Nu Q[W]
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1.00 1.299E+03 1.900 0.4530
9.71 1.261E+04 3.354 7.7628
18.42 2.393E+04 3.936 17.2824
27.12 3.524E+04 4.336 28.0415
35.83 4.656E+04 4.649 39.7145
44.54 5.787E+04 4.909 52.1252
53.25 6.918E+04 5.133 65.1612
61.96 8.050E+04 5.331 78.7434
70.67 9.181E+04 5.509 92.8128
79.38 1.031E+05 5.672 107.3234
88.08 1.144E+05 5.821 122.2381
96.79 1.258E+05 5.960 137.5267
105.50 1.371E+05 6.090 153.1634
114.21 1.484E+05 6.212 169.1266
122.92 1.597E+05 6.327 185.3971
131.62 1.710E+05 6.436 201.9587
140.33 1.823E+05 6.540 218.7966
149.04 1.936E+05 6.639 235.8980
157.75 2.050E+05 6.734 253.2511
166.46 2.163E+05 6.825 270.8454
175.17 2.276E+05 6.913 288.6715
183.88 2.389E+05 6.997 306.7206
192.58 2.502E+05 7.078 324.9847
201.29 2.615E+05 7.157 343.4566
210.00 2.728E+05 7.233 362.1293
--- CORRELATIONS ---
Vertical rect: Catton (1978) correlations
Horizontal hot bottom: Globe-Dropkin Nu=0.069*Ra^(1/3)*Pr^0.074
Concentric cyl: k_eff/k=0.386(Pr/(0.861+Pr))^0.25*Ra^0.25
Concentric sph: k_eff/k=0.74(Pr/(0.861+Pr))^0.25*Ra^0.25
Ref: Incropera Ch.9 Sec.9.8, Kothandaraman Ch.10
Calculation Methodology
Mathematical Model & Theory
Natural convection in enclosures is governed by Rayleigh number $\text{Ra}_L = g \beta \Delta T L^3 / (\nu \alpha)$. Different correlations apply:
- Vertical Rectangular Cavities: Catton's correlations based on aspect ratio $H/L$:
$$\text{Nu} = C(Pr) \text{Ra}^{0.28-0.29} (H/L)^{-0.25-0.30}$$
- Horizontal Cavities (Hot Bottom): Globe-Dropkin correlation:
$$\text{Nu} = 0.069 \text{Ra}^{1/3} \text{Pr}^{0.074}$$
- Concentric Annulus: Raithby & Hollands formulation:
$$\frac{k_{eff}}{k} = 0.386 \left(\frac{\text{Pr}}{0.861 + \text{Pr}}\right)^{1/4} \text{Ra}_L^{1/4}$$
Academic References:
- Incropera, F. P., & DeWitt, D. P. (2011). Fundamentals of Heat and Mass Transfer. 7th Edition, John Wiley & Sons.
- Catton, I. (1978). Natural Convection in Enclosures. Proceedings of the 6th International Heat Transfer Conference.
Worked Engineering Example
A vertical double-pane window has a gap $L = 20\text{ mm}$ and height $H = 0.5\text{ m}$. The inner glass is at $60^\circ\text{C}$ and the outer glass is at $20^\circ\text{C}$. Find the heat transfer coefficient of the air slot ($T_f = 40^\circ\text{C}$).
Step-by-step Solution:
1. Evaluate properties at $40^\circ\text{C}$:
- Air: $\nu = 1.7 \times 10^{-5}\text{ m}^2\text{/s}$, $\alpha = 2.4 \times 10^{-5}\text{ m}^2\text{/s}$, $Pr = 0.707$, $\beta = 1/313.15 = 0.00319\text{ K}^{-1}$, $k = 0.0271\text{ W/m·K}$.
2. Calculate Rayleigh number $\text{Ra}_L$:
$$\text{Ra}_L = \frac{9.81 \times 0.00319 \times (60-20) \times 0.02^3}{1.7 \times 10^{-5} \times 2.4 \times 10^{-5}} = 24,534$$ 3. Evaluate aspect ratio and choose Catton correlation ($H/L = 0.5/0.02 = 25$):
Since $H/L > 10$, we use: $$\text{Nu} = 0.42 \text{Ra}_L^{0.25} \text{Pr}^{0.012} (H/L)^{-0.3}$$ $$\text{Nu} = 0.42 \times (24,534)^{0.25} \times (0.707)^{0.012} \times (25)^{-0.3} = 0.42 \times 12.52 \times 0.996 \times 0.38 = 1.99$$ 4. Convection coefficient $h_{eff}$:
$$h_{eff} = \frac{\text{Nu} k}{L} = \frac{1.99 \times 0.0271}{0.02} = 2.70\text{ W/m}^2\text{K}$$