Training Convection & Dimensionless Numbers Natural Convection and Grashof/Rayleigh Numbers
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Natural Convection and Grashof/Rayleigh Numbers

24 min Convection & Dimensionless Numbers

Natural Convection and Grashof/Rayleigh Numbers

When there is no fan or pump, buoyancy drives the flow. Hot fluid rises, cold fluid sinks, and the resulting circulation is called natural (or free) convection. The Grashof and Rayleigh numbers are the buoyancy equivalents of the Reynolds number.

Grashof Number

$$Gr = \frac{g\beta(T_s - T_\infty)L_c^3}{\nu^2}$$

Ratio of buoyancy to viscous forces. $g = 9.81$ m/s², $\beta$ = volumetric thermal expansion coefficient (for ideal gas: $\beta = 1/T$ in Kelvin), $L_c$ = characteristic length.

Rayleigh Number

$$Ra = Gr \cdot Pr = \frac{g\beta(T_s - T_\infty)L_c^3}{\nu \alpha}$$

Transition from laminar to turbulent natural convection: $Ra \approx 10^9$.

Churchill–Chu Correlation (Vertical Plate)

For all $Ra$ on a vertical plate:

$$Nu = \left[0.825 + \frac{0.387\,Ra^{1/6}}{\left(1 + (0.492/Pr)^{9/16}\right)^{8/27}}\right]^2$$

For laminar only ($Ra < 10^9$), a simpler form:

$$Nu = 0.68 + \frac{0.670\,Ra^{1/4}}{\left(1 + (0.492/Pr)^{9/16}\right)^{4/9}}$$

Example 1 — Heated Vertical Wall

A 1.5 m tall wall at 60°C in 20°C air. Properties at film temp (40°C): $\nu = 1.7 \times 10^{-5}$, $\alpha = 2.4 \times 10^{-5}$, $Pr = 0.71$, $k = 0.027$, $\beta = 1/313$ K⁻¹. Find $h$.

$Ra = \frac{9.81 \times (1/313) \times 40 \times 1.5^3}{1.7 \times 10^{-5} \times 2.4 \times 10^{-5}} = \frac{9.81 \times 0.1278 \times 3.375}{4.08 \times 10^{-10}}$

$= \frac{4.231}{4.08 \times 10^{-10}} = 1.037 \times 10^{10}$

$Ra > 10^9$ → turbulent. Use full Churchill–Chu:

$\psi = (1 + (0.492/0.71)^{9/16})^{8/27} = (1 + 0.626)^{8/27} = 1.626^{0.296} = 1.161$

$Nu = [0.825 + 0.387 \times (1.037 \times 10^{10})^{1/6}/1.161]^2$

$(1.037 \times 10^{10})^{1/6} = 10^{10/6} \times 1.037^{1/6} = 4642 \times 1.006 = 4670$

$Nu = [0.825 + 0.387 \times 4670/1.161]^2 = [0.825 + 1557]^2 \approx 1558^2$

That seems high — let’s recheck: $(10^{10})^{1/6} = 10^{1.667} = 46.42$, so $(1.037 \times 10^{10})^{1/6} = 46.7$

$Nu = [0.825 + 0.387 \times 46.7/1.161]^2 = [0.825 + 15.57]^2 = [16.4]^2 = 269$

$$h = \frac{269 \times 0.027}{1.5} = 4.84 \text{ W/(m²\cdot K)}$$

Typical for natural convection in air.

Example 2 — Hot Horizontal Pipe

A 5 cm diameter horizontal pipe at 90°C in 25°C air ($Ra_D = 1.2 \times 10^5$, $k = 0.028$). Use the correlation $Nu = 0.36 + 0.518\,Ra^{0.25}$. Find $h$.

$Nu = 0.36 + 0.518 \times (1.2 \times 10^5)^{0.25} = 0.36 + 0.518 \times 18.6 = 0.36 + 9.63 = 9.99$

$h = 9.99 \times 0.028 / 0.05 = 5.60$ W/(m²·K)

Example 3 — Forced vs. Natural

If $Gr/Re^2 \gg 1$, which regime dominates?

$Gr/Re^2 \gg 1$ means buoyancy forces dominate inertial forces → natural convection dominates.

$Gr/Re^2 \ll 1$ → forced convection dominates. $Gr/Re^2 \approx 1$ → mixed convection.

Practice Problems

1. Find $Gr$ for a 0.5 m plate, $T_s = 100$°C, $T_\infty = 30$°C, $\beta = 1/338$, $\nu = 1.8 \times 10^{-5}$.
2. Is the flow in Problem 1 laminar or turbulent? ($Pr = 0.71$)
3. Air at $T_\infty = 20$°C, $T_s = 50$°C. Film temp? Why use film temp for properties?
4. For an ideal gas at 400 K, find $\beta$.
5. $Gr = 10^8$, $Re = 5000$. Is this forced, natural, or mixed?
6. Doubling $L_c$ changes $Gr$ by what factor?
Show Answer Key

1. $Gr = 9.81 \times (1/338) \times 70 \times 0.5^3 / (1.8 \times 10^{-5})^2 = 9.81 \times 0.207 \times 0.125 / (3.24 \times 10^{-10}) = 0.2537/3.24 \times 10^{-10} = 7.83 \times 10^8$

2. $Ra = Gr \times Pr = 7.83 \times 10^8 \times 0.71 = 5.56 \times 10^8 < 10^9$ → Laminar

3. $T_f = (20 + 50)/2 = 35$°C. Film temp gives better average properties across the boundary layer.

4. $\beta = 1/T = 1/400 = 0.0025$ K⁻¹

5. $Gr/Re^2 = 10^8/(5000)^2 = 4.0 \approx 1$ → Mixed convection

6. Factor of $2^3 = 8$ ($Gr \propto L_c^3$)

Nusselt Number and Forced Convection Correlations Practice Test — Convection & Dimensionless Numbers