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 = 1.5905E+04
Whitaker Nu = 76.1093
Ranz Nu = 69.5061
Coeff h = 40.0335 W/m2K
Transfer Q = 23.5817 W
--- VELOCITY SWEEP ---
V[m/s] Re Nu_Whitaker Nu_Ranz h[W/m2K] Q[W]
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0.050 1.591E+02 7.706 8.751 4.0534 2.3877
0.881 2.803E+03 29.757 30.341 15.6523 9.2200
1.712 5.448E+03 42.326 41.507 22.2636 13.1144
2.544 8.092E+03 52.442 50.150 27.5847 16.2487
3.375 1.074E+04 61.230 57.462 32.2068 18.9713
4.206 1.338E+04 69.144 63.916 36.3698 21.4236
5.037 1.602E+04 76.426 69.759 40.2003 23.6799
5.869 1.867E+04 83.223 75.136 43.7750 25.7856
6.700 2.131E+04 89.629 80.144 47.1447 27.7705
7.531 2.396E+04 95.713 84.850 50.3448 29.6556
8.363 2.660E+04 101.524 89.302 53.4018 31.4563
9.194 2.925E+04 107.102 93.539 56.3356 33.1844
10.025 3.189E+04 112.475 97.587 59.1619 34.8492
10.856 3.453E+04 117.668 101.471 61.8932 36.4581
11.688 3.718E+04 122.699 105.209 64.5397 38.0170
12.519 3.982E+04 127.586 108.817 67.1100 39.5310
13.350 4.247E+04 132.341 112.306 69.6111 41.0043
14.181 4.511E+04 136.976 115.688 72.0492 42.4405
15.012 4.776E+04 141.501 118.973 74.4295 43.8426
15.844 5.040E+04 145.925 122.168 76.7564 45.2133
16.675 5.304E+04 150.255 125.280 79.0340 46.5549
17.506 5.569E+04 154.497 128.315 81.2656 47.8694
18.337 5.833E+04 158.659 131.279 83.4545 49.1587
19.169 6.098E+04 162.744 134.177 85.6032 50.4244
20.000 6.362E+04 166.757 137.012 87.7143 51.6680
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:
$$Q = h A_s (T_s - T_\infty) = 39.9 \times 0.00785 \times (100-25) = 23.5\text{ W}$$