Forced Convection Over Spheres
Determine boundary layer heat transfer coefficients and total heat rate for flows over spherical surfaces.
External Flow Over Sphere
Similar to crossflow over cylinders, flow separates from the rear side of spheres, creating a low-pressure wake region. Convection is analyzed using empirical correlations that account for the fluid viscosity change near the surface ($\mu/\mu_s$).
Parameters Setup
Results & Curves
📈 Nusselt number vs Velocity
Engine Output
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CONVECTION OVER SPHERE ENGINE
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Reynolds Re = 1.2724E+04
Whitaker Nu = 68.3574
Ranz Nu = 62.3793
Coeff h = 179.7799 W/m2K
Transfer Q = 2.2592 W
--- VELOCITY SWEEP ---
V[m/s] Re Nu_Whitaker Nu_Ranz h[W/m2K] Q[W]
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0.050 3.181E+01 4.437 5.019 11.6701 0.1467
3.381 2.151E+03 26.345 26.826 69.2865 0.8707
6.713 4.271E+03 37.755 36.980 99.2968 1.2478
10.044 6.390E+03 46.881 44.788 123.2977 1.5494
13.375 8.509E+03 54.786 51.376 144.0878 1.8107
16.706 1.063E+04 61.895 57.184 162.7829 2.0456
20.038 1.275E+04 68.428 62.436 179.9664 2.2615
23.369 1.487E+04 74.521 67.267 195.9897 2.4629
26.700 1.699E+04 80.260 71.764 211.0844 2.6526
30.031 1.911E+04 85.708 75.988 225.4130 2.8326
33.362 2.123E+04 90.911 79.983 239.0950 3.0046
36.694 2.335E+04 95.902 83.784 252.2212 3.1695
40.025 2.546E+04 100.708 87.416 264.8624 3.3284
43.356 2.758E+04 105.352 90.899 277.0756 3.4818
46.687 2.970E+04 109.851 94.251 288.9072 3.6305
50.019 3.182E+04 114.219 97.486 300.3955 3.7749
53.350 3.394E+04 118.469 100.614 311.5727 3.9153
56.681 3.606E+04 122.611 103.647 322.4664 4.0522
60.013 3.818E+04 126.654 106.591 333.0999 4.1859
63.344 4.030E+04 130.606 109.455 343.4938 4.3165
66.675 4.242E+04 134.474 112.244 353.6657 4.4443
70.006 4.454E+04 138.263 114.964 363.6315 4.5695
73.338 4.666E+04 141.979 117.621 373.4048 4.6923
76.669 4.878E+04 145.627 120.218 382.9982 4.8129
80.000 5.090E+04 149.210 122.759 392.4226 4.9313
Calculation Methodology
Mathematical Model & Theory
Convective heat transfer over a spherical object is evaluated using two primary models:
1. Whitaker Correlation:
$$\text{Nu}_D = 2 + \left(0.4 \text{Re}_D^{1/2} + 0.06 \text{Re}_D^{2/3}\right) \text{Pr}^{0.4} \left(\frac{\mu}{\mu_s}\right)^{1/4}$$
where fluid properties (except $\mu_s$) are evaluated at the free-stream temperature $T_\infty$.
2. Ranz-Marshall Correlation:
$$\text{Nu}_D = 2 + 0.6 \text{Re}_D^{1/2} \text{Pr}^{1/3}$$
This correlation is a classical simplification widely used for droplets and evaporating sprays.
Academic References:
- Incropera, F. P., & DeWitt, D. P. (2011). Fundamentals of Heat and Mass Transfer. 7th Edition, John Wiley & Sons.
- Whitaker, S. (1972). Forced Convection Heat Transfer Correlations for Flow in Pipes, Past Flat Plates, Single Cylinders, Single Spheres, and for Flow in Packed Beds and Tube Bundles. AIChE J.
Worked Engineering Example
Air at $25^\circ\text{C}$ flows at $5.0\text{ m/s}$ past a $50\text{ mm}$ sphere maintained at $100^\circ\text{C}$. Find the convective heat loss ($T_\infty = 25^\circ\text{C}$).
Step-by-step Solution:
1. Evaluate air properties at free-stream temp $25^\circ\text{C}$:
- $\rho = 1.177\text{ kg/m}^3, \mu = 1.85 \times 10^{-5}\text{ Pa·s}, k = 0.0263\text{ W/m·K}, Pr = 0.71$.
- At $T_s = 100^\circ\text{C}$, $\mu_s \approx 2.18 \times 10^{-5}\text{ Pa·s}$.
2. Calculate Reynolds number:
$$\text{Re}_D = \frac{\rho V D}{\mu} = \frac{1.177 \times 5.0 \times 0.05}{1.85 \times 10^{-5}} = 15,905$$ 3. Evaluate Nusselt number via Whitaker correlation:
$$\text{Nu}_D = 2 + \left[0.4 \times (15,905)^{1/2} + 0.06 \times (15,905)^{2/3}\right] \times (0.71)^{0.4} \times \left(\frac{1.85}{2.18}\right)^{1/4}$$ $$\text{Nu}_D = 2 + \left[50.4 + 37.9\right] \times 0.87 \times 0.96 = 75.8$$ 4. Convection coefficient $h$:
$$h = \frac{\text{Nu}_D k}{D} = \frac{75.8 \times 0.0263}{0.05} = 39.9\text{ W/m}^2\text{K}$$ 5. Surface area $A_s = \pi D^2 = \pi \times 0.05^2 = 0.00785\text{ m}^2$. Total heat loss:
$$Q = h A_s (T_s - T_\infty) = 39.9 \times 0.00785 \times (100-25) = 23.5\text{ W}$$