Chapter 9: Free Convection

Source files used: Heat Transfer (31), (32), (33), and combined-mode examples in (34), (35); textbook Chapter 9 as support. No worked examples are included.

1. Big Picture

Chapter 9 is convection without a fan or pump.

In forced convection, fluid motion is imposed by something external: a fan, pump, vehicle speed, wind, etc. In free convection, the fluid moves because of buoyancy. Temperature differences create density differences; in a gravitational field, those density differences create motion.

Biddle’s wall example captures the physics:

• A warm wall heats the nearby air.
• Warm air becomes less dense.
• Less dense air rises.
• That rising motion creates convection heat transfer.

So Chapter 9 still asks for $h$, but now the velocity is not given directly. The motion is caused by buoyancy.

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2. Core Ideas

2.1 Free vs Forced Convection

Forced convection:

fluid motion caused by fan, pump, wind, or imposed velocity

Free convection:

fluid motion caused by density differences and gravity

Free convection velocities are usually smaller than forced convection velocities, so free convection heat transfer coefficients are usually smaller.

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2.2 Buoyancy and Density Gradients

For gases, density changes with temperature. Warm air is less dense; cold air is more dense.

If a vertical wall is hotter than the surrounding air:

• air near the wall warms,
• density decreases,
• buoyancy drives upward flow.

If a vertical wall is colder than the surrounding air:

• air near the wall cools,
• density increases,
• fluid tends to move downward.

The flow direction matters for physical interpretation, but the heat-transfer correlations usually use $|T_s-T_\infty|$.

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2.3 Grashof and Rayleigh Numbers

In forced convection, Reynolds number measures inertial effects from imposed velocity. In free convection, there is no imposed velocity, so the key group is the Grashof number:

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

Most free-convection correlations use Rayleigh number:

$$Ra_L=Gr_LPr$$

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2.4 Geometry Matters

Free convection correlations depend heavily on geometry and orientation:

• vertical plate,
• vertical cylinder,
• horizontal plate,
• horizontal cylinder.

For horizontal plates, you must know whether the hot surface faces upward or downward. That changes whether the buoyancy motion is stable or unstable.

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2.5 Combined Free Convection and Radiation

Biddle’s combined-mode examples often use:

$$\dot{q}=\dot{q}_{\text{conv}}+\dot{q}_{\text{rad}}$$ $$q'=\bar h\pi D(T_s-T_\infty)+\varepsilon\sigma\pi D(T_s^4-T_{\text{sur}}^4)$$

Radiation can be comparable to free convection because free convection $h$ may be small.

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3. Main Governing Equations and Formulas

3.1 Film Temperature

$$T_f=\frac{T_s+T_\infty}{2}$$

Use properties at $T_f$ unless specified otherwise.

For ideal gases:

$$\beta=\frac{1}{T_f}$$

where $T_f$ is in K.

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3.2 Grashof Number

$$Gr_L=\frac{g\beta|T_s-T_\infty|L^3}{\nu^2}$$

where:

• $g$: gravitational acceleration,
• $\beta$: volumetric thermal expansion coefficient,
• $L$: characteristic length,
• $\nu$: kinematic viscosity.

Use for free convection.

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3.3 Rayleigh Number

$$Ra_L=Gr_LPr$$

Most Chapter 9 correlations are functions of $Ra$.

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3.4 Nusselt Number

$$Nu_L=\frac{\bar hL}{k_f}$$

or

$$Nu_D=\frac{\bar hD}{k_f}$$

Then:

$$\bar h=\frac{Nu\,k_f}{L_c}$$

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3.5 Vertical Plate — Churchill-Chu Correlation

For a vertical plate, a widely used all-range correlation is:

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

Use for free convection over a vertical plate when conditions fall within the correlation validity range.

Characteristic length:

$$L=\text{vertical height}$$

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3.6 Vertical Cylinder

A vertical cylinder may be treated like a vertical plate if its diameter is large enough relative to the boundary-layer thickness. A common criterion is:

$$\frac{D}{L}\gtrsim \frac{35}{Gr_L^{1/4}}$$

If this criterion is satisfied, use vertical plate correlations. If not, use a vertical-cylinder-specific correlation from the table.

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3.7 Horizontal Cylinder — Churchill-Chu Correlation

For a horizontal cylinder:

$$\overline{Nu}_D=\left[0.60+\frac{0.387Ra_D^{1/6}}{\left(1+(0.559/Pr)^{9/16}\right)^{8/27}}\right]^2$$

where:

$$Ra_D=Gr_DPr$$

and characteristic length is diameter $D$.

Use for pipes/tubes in free convection.

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3.8 Horizontal Plates: Characteristic Length

For horizontal plates:

$$L_c=\frac{A_s}{P}$$

where:

• $A_s$: surface area,
• $P$: perimeter of the plate.

This is not necessarily the plate length.

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3.9 Horizontal Plate Correlations

For a hot surface facing upward or cold surface facing downward:

$$\overline{Nu}_L=0.54Ra_L^{1/4}\qquad (10^4\lesssim Ra_L\lesssim10^7)$$ $$\overline{Nu}_L=0.15Ra_L^{1/3}\qquad (10^7\lesssim Ra_L\lesssim10^{11})$$

For a hot surface facing downward or cold surface facing upward:

$$\overline{Nu}_L=0.27Ra_L^{1/4}$$

Use the orientation rule carefully. It is a common exam trap.

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3.10 Free Convection Heat Transfer Rate

$$\dot{q}=\bar hA_s(T_s-T_\infty)$$

Use after $\bar h$ is found from the correct free-convection correlation.

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3.11 Radiation with Free Convection

For a small surface/object in large surroundings:

$$\dot{q}_{\text{rad}}=\varepsilon\sigma A_s(T_s^4-T_{\text{sur}}^4)$$

Total heat transfer:

$$\dot{q}_{\text{total}}=\dot{q}_{\text{conv}}+\dot{q}_{\text{rad}}$$

For a horizontal cylinder per unit length:

$$q'_{\text{conv}}=\bar h\pi D(T_s-T_\infty)$$ $$q'_{\text{rad}}=\varepsilon\sigma\pi D(T_s^4-T_{\text{sur}}^4)$$ $$q'_{\text{total}}=q'_{\text{conv}}+q'_{\text{rad}}$$

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4. Problem-Solving Workflow

1. Identify free convection: no imposed velocity; motion caused by buoyancy.
2. Identify geometry: vertical plate, vertical cylinder, horizontal plate, horizontal cylinder.
3. Determine characteristic length.
4. Compute film temperature $T_f$.
5. Evaluate fluid properties at $T_f$.
6. For ideal gas, compute $\beta=1/T_f$.
7. Compute $Gr$ and $Ra=GrPr$.
8. Select the correct correlation based on geometry and orientation.
9. Compute $Nu$, then $\bar h=Nu k/L_c$.
10. Compute $\dot{q}_{\text{conv}}$.
11. If radiation is present, compute $\dot{q}_{\text{rad}}$ and add it.

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5. Decision Rules / Decision Trees

5.1 Forced or Free?

Fan, pump, wind, imposed velocity? → forced convection

No imposed velocity, motion caused by hot/cold density differences? → free convection

5.2 Geometry Selection

Vertical flat wall? → vertical plate correlation

Vertical cylinder? → check if vertical plate approximation is allowed

Horizontal pipe/tube? → horizontal cylinder correlation

Horizontal flat surface? → horizontal plate correlation with orientation check

5.3 Horizontal Plate Orientation

Hot surface facing upward? → unstable buoyancy, stronger convection

Hot surface facing downward? → stable stratification, weaker convection

Cold surface facing downward? → unstable buoyancy, stronger convection

Cold surface facing upward? → stable stratification, weaker convection

5.4 Radiation Inclusion

Free convection $h$ is small? AND surface temperature is significantly different from surroundings? AND emissivity is not tiny? → radiation may be important

Problem gives $\varepsilon$ and $T_{\text{sur}}$? → include radiation unless told to neglect it

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6. Important Tables / Correlations Needed

6.1 Free Convection Correlation Summary

Geometry	Characteristic Length	Correlation Type
Vertical plate	height $L$	Churchill-Chu vertical plate
Vertical cylinder	height $L$, check $D/L$ criterion	vertical plate approximation or cylinder table
Horizontal cylinder	diameter $D$	Churchill-Chu horizontal cylinder
Horizontal plate	$A_s/P$	orientation-dependent plate correlations

6.2 Horizontal Plate Orientation Summary

Surface Condition	Buoyancy Behavior	Correlation Level
Hot surface facing up	unstable	stronger convection
Hot surface facing down	stable	weaker convection
Cold surface facing down	unstable	stronger convection
Cold surface facing up	stable	weaker convection

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7. Key Takeaways

• Free convection is buoyancy-driven flow.
• It usually gives lower $h$ than forced convection.
• Use $Gr$ and $Ra$, not Reynolds number, as the main free-convection groups.
• Always evaluate properties at film temperature unless told otherwise.
• For gases, $\beta=1/T_f$ with $T_f$ in K.
• Geometry and orientation strongly affect the correlation.
• Horizontal plate problems require careful “hot side up/down” logic.
• Radiation often matters in free convection because $h$ is relatively small.
• Combined-mode problems often use $\dot{q}=\dot{q}_{\text{conv}}+\dot{q}_{\text{rad}}$.