Combined Free & Forced Convection
Analyze heat transfer in mixed convection regimes using Richardson number criteria.
Mixed Convection Regime
Combined convection analysis applies when both inertial (forced) forces and buoyancy (natural) forces play significant roles. The Richardson number is defined as:
$$\text{Ri} = \frac{\text{Gr}}{\text{Re}^2}$$
- $\text{Ri} \ll 0.1$: Forced convection dominates.
- $\text{Ri} \gg 10$: Natural convection dominates.
- $0.1 \le \text{Ri} \le 10$: Mixed convection regime.
Parameters Setup
Results & Curves
📈 Nusselt number vs Velocity
Engine Terminal Output
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COMBINED FREE & FORCED CONVECTION ENGINE
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Reynolds Re = 6.3622E+03
Grashof Number = 4.4014E+09
Richardson Number Ri = 1.0874E+02
Nu Forced = 47.2487
Nu Natural = 175.0225
Nu Combined = 176.1628
Coeff h = 4.6331 W/m2K
--- VELOCITY SWEEP ---
V[m/s] Re Gr Ri Nu_f Nu_n Nu_c h[W/m2K]
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0.010 3.181E+01 4.401E+09 4.350E+06 3.34 175.02 175.02 4.603
0.260 8.258E+02 4.401E+09 6.455E+03 17.02 175.02 175.08 4.605
0.509 1.620E+03 4.401E+09 1.678E+03 23.84 175.02 175.17 4.607
0.759 2.414E+03 4.401E+09 7.555E+02 29.10 175.02 175.29 4.610
1.008 3.208E+03 4.401E+09 4.278E+02 33.55 175.02 175.43 4.614
1.258 4.002E+03 4.401E+09 2.749E+02 37.47 175.02 175.59 4.618
1.507 4.795E+03 4.401E+09 1.914E+02 41.02 175.02 175.77 4.623
1.757 5.589E+03 4.401E+09 1.409E+02 44.29 175.02 175.96 4.628
2.007 6.383E+03 4.401E+09 1.080E+02 47.33 175.02 176.17 4.633
2.256 7.177E+03 4.401E+09 8.544E+01 50.18 175.02 176.39 4.639
2.506 7.971E+03 4.401E+09 6.927E+01 52.89 175.02 176.62 4.645
2.755 8.765E+03 4.401E+09 5.729E+01 55.46 175.02 176.86 4.651
3.005 9.559E+03 4.401E+09 4.817E+01 57.92 175.02 177.11 4.658
3.255 1.035E+04 4.401E+09 4.106E+01 60.27 175.02 177.37 4.665
3.504 1.115E+04 4.401E+09 3.542E+01 62.54 175.02 177.64 4.672
3.754 1.194E+04 4.401E+09 3.087E+01 64.73 175.02 177.93 4.679
4.003 1.273E+04 4.401E+09 2.714E+01 66.85 175.02 178.21 4.687
4.253 1.353E+04 4.401E+09 2.405E+01 68.90 175.02 178.51 4.695
4.503 1.432E+04 4.401E+09 2.146E+01 70.89 175.02 178.82 4.703
4.752 1.512E+04 4.401E+09 1.926E+01 72.83 175.02 179.13 4.711
5.002 1.591E+04 4.401E+09 1.739E+01 74.72 175.02 179.45 4.720
5.251 1.670E+04 4.401E+09 1.577E+01 76.56 175.02 179.78 4.728
5.501 1.750E+04 4.401E+09 1.437E+01 78.36 175.02 180.11 4.737
5.750 1.829E+04 4.401E+09 1.315E+01 80.12 175.02 180.45 4.746
6.000 1.909E+04 4.401E+09 1.208E+01 81.84 175.02 180.79 4.755
Calculation Methodology
Mathematical Model & Theory
In mixed convection, the Nusselt numbers for forced ($\text{Nu}_f$) and natural ($\text{Nu}_n$) convection are computed separately and then superposed using a power-law exponent of $3$ (Incropera Ch. 9): $$\text{Nu}_c^3 = \text{Nu}_f^3 \pm \text{Nu}_n^3$$ - Assisting Flow ($+$): Buoyancy acts in the direction of the forced flow, enhancing heat transfer. $$\text{Nu}_c = \left(\text{Nu}_f^3 + \text{Nu}_n^3\right)^{1/3}$$ - Opposing Flow ($-$): Buoyancy acts against the forced flow, creating turbulent shear layers or retarding flow. $$\text{Nu}_c = \left|\text{Nu}_f^3 - \text{Nu}_n^3\right|^{1/3}$$ - Transverse Flow ($+$): Buoyancy acts perpendicular to forced flow, inducing secondary cross-flow currents. $$\text{Nu}_c = \left(\text{Nu}_f^3 + \text{Nu}_n^3\right)^{1/3}$$
Academic References:
- Incropera, F. P., & DeWitt, D. P. (2011). Fundamentals of Heat and Mass Transfer. 7th Edition, John Wiley & Sons.
- Churchill, S. W. (1977). A Comprehensive Correlating Equation for Laminar, Assisting, Natural Convection on a Vertical Plate.
Worked Engineering Example
Air flows upwards at $0.5\text{ m/s}$ along a vertical flat plate of length $0.5\text{ m}$ at a surface temperature of $100^\circ\text{C}$ in assisting flow, with ambient air at $25^\circ\text{C}$. Calculate the combined convection coefficient.
Step-by-step Solution:
1. Evaluate properties at film temp $T_f = (100+25)/2 = 62.5^\circ\text{C}$:
- Air $\nu \approx 1.95 \times 10^{-5}\text{ m}^2\text{/s}$, $\beta \approx 1/(62.5+273.15) = 0.00298\text{ K}^{-1}$, $k_f = 0.028\text{ W/mK}$, $Pr = 0.7$.
2. Calculate Reynolds and Grashof numbers:
$$\text{Re} = \frac{V L_c}{\nu} = \frac{0.5 \times 0.5}{1.95 \times 10^{-5}} = 12,820$$ $$\text{Gr} = \frac{g \beta (T_s - T_\infty) L_c^3}{\nu^2} = \frac{9.81 \times 0.00298 \times (100-25) \times 0.5^3}{(1.95 \times 10^{-5})^2} = 7.2 \times 10^8$$ 3. Evaluate Richardson number:
$$\text{Ri} = \frac{\text{Gr}}{\text{Re}^2} = \frac{7.2 \times 10^8}{12,820^2} = 4.38$$ Since $0.1 < \text{Ri} < 10$, mixed convection governs the flow.
4. Compute Nusselt numbers and combine:
- Forced: $\text{Nu}_f \approx 67.5$, Natural: $\text{Nu}_n \approx 94.2$
- Combined: $\text{Nu}_c = (67.5^3 + 94.2^3)^{1/3} = 104.6$
5. Calculate heat transfer coefficient $h$:
$$h = \frac{\text{Nu}_c k_f}{L_c} = \frac{104.6 \times 0.028}{0.5} = 5.86\text{ W/m}^2\text{K}$$