3.4 Natural Convection

Natural convection principles

Natural convection is heat transfer driven by buoyancy forces rather than external devices like fans or pumps. When a fluid near a surface gets heated, its density decreases, causing it to rise while cooler, denser fluid sinks to take its place. This density-driven circulation transfers heat without any mechanical forcing.

Understanding natural convection matters because it governs heat transfer in situations where no forced flow exists: cooling of electronics, heating in buildings, and heat loss from industrial equipment.

Buoyancy-driven fluid motion

The driving mechanism behind natural convection is the buoyancy force, which arises from density differences between warmer and cooler regions of a fluid under the influence of gravity.

Here's how the process works:

1. A surface at temperature $T_s$ heats the adjacent fluid above the bulk fluid temperature $T_\infty$.
2. The heated fluid expands and becomes less dense than the surrounding cooler fluid.
3. Gravity acts on this density difference, pushing the lighter (warmer) fluid upward and pulling the heavier (cooler) fluid downward.
4. This creates a circulation pattern known as a convection current.

The strength of the buoyancy force depends on:

• The temperature difference $\Delta T = T_s - T_\infty$ between the heated surface and the surrounding fluid
• The fluid's volumetric thermal expansion coefficient $\beta$, which quantifies how much the fluid's density changes per degree of temperature change
• Gravitational acceleration $g$
• Fluid properties like density $\rho$ and kinematic viscosity $\nu$

Boundary layer development

Just like in forced convection, boundary layers form near the heated surface in natural convection.

• A velocity boundary layer develops because the fluid velocity is zero at the surface (no-slip condition) and increases to some maximum before decaying back to zero far from the surface. This profile is different from forced convection, where velocity approaches the free-stream value.
• A thermal boundary layer develops because the fluid temperature transitions from $T_s$ at the surface to $T_\infty$ in the bulk fluid.

The thickness of both boundary layers depends on fluid properties, surface geometry, and the strength of the buoyancy force. For example, a heated vertical plate in still air develops both layers simultaneously: the air nearest the plate heats up, rises along the surface, and entrains cooler air from the surroundings.

Heat transfer over surfaces

Vertical surfaces

For a vertical heated surface, the buoyancy force acts parallel to the surface. Heated fluid rises along the plate, forming a boundary layer that grows thicker with distance from the leading edge (bottom of the plate).

• The local heat transfer coefficient is highest near the bottom of the plate (where the boundary layer is thinnest) and decreases with height.
• The average heat transfer coefficient depends on the plate height $L$, the temperature difference $\Delta T$, and fluid properties.
• The widely used Churchill and Chu correlation provides the average Nusselt number for vertical surfaces (covered in the correlations section below).

Horizontal and inclined surfaces

Horizontal surfaces behave quite differently from vertical ones because the buoyancy force acts perpendicular to the surface. The heat transfer depends heavily on the heating configuration:

• Hot side facing up (or cold side facing down): This is an unstable configuration. The lighter hot fluid is trapped beneath cooler, denser fluid, so it tends to break through in a pattern of rising plumes called Bénard cells. This promotes mixing and enhances heat transfer.
• Hot side facing down (or cold side facing up): This is a stable configuration. The hot, lighter fluid is already on top, so there's little driving force for circulation. Heat transfer is suppressed and relies more on conduction.

For inclined surfaces, the buoyancy force has components both parallel and perpendicular to the surface. The behavior transitions between vertical-surface and horizontal-surface characteristics depending on the inclination angle $\theta$. As a practical example, a tilted solar collector absorber plate experiences natural convection with the effective buoyancy component along the surface reduced by a factor of $\cos(\theta)$.

Rayleigh number for flow regime

Dimensionless parameter

The Rayleigh number ($Ra$) is the key dimensionless parameter in natural convection. It characterizes the relative strength of buoyancy-driven flow compared to viscous and thermal damping effects.

$Ra$ is defined as the product of the Grashof number ($Gr$) and the Prandtl number ($Pr$):

$Ra = Gr \cdot Pr = \frac{g \beta \Delta T L^3}{\nu \alpha}$

where:

• $g$ = gravitational acceleration
• $\beta$ = volumetric thermal expansion coefficient (for an ideal gas, $\beta = 1/T_f$ where $T_f$ is the film temperature in Kelvin)
• $\Delta T = T_s - T_\infty$
• $L$ = characteristic length of the surface
• $\nu$ = kinematic viscosity
• $\alpha$ = thermal diffusivity

The Grashof number $Gr = \frac{g \beta \Delta T L^3}{\nu^2}$ represents the ratio of buoyancy forces to viscous forces. Think of it as the natural convection equivalent of the Reynolds number in forced convection.

The Prandtl number $Pr = \frac{\nu}{\alpha}$ represents the ratio of momentum diffusivity to thermal diffusivity.

Flow regime determination

The magnitude of $Ra$ tells you whether the flow is laminar, transitional, or turbulent:

For vertical surfaces, the critical Rayleigh number marking the transition from laminar to turbulent flow is approximately $Ra_{cr} \approx 10^9$.

For horizontal surfaces heated from below, the critical Rayleigh number for the onset of convection (where buoyancy first overcomes viscous resistance) is much lower, around $Ra_{cr} \approx 1708$ for a fluid layer confined between two horizontal plates.

At low $Ra$, heat transfer is dominated by conduction with gentle laminar motion. At high $Ra$, turbulent mixing significantly enhances heat transfer rates.

Empirical correlations for heat transfer

Nusselt number and heat transfer coefficient

Empirical correlations express the average Nusselt number $\overline{Nu}$ as a function of $Ra$ and $Pr$ for a given surface geometry. The Nusselt number represents the ratio of convective to conductive heat transfer across the boundary layer.

Once you have $\overline{Nu}$, you find the average heat transfer coefficient from:

$h = \frac{\overline{Nu} \cdot k}{L}$

where $k$ is the fluid's thermal conductivity and $L$ is the characteristic length.

All fluid properties in these correlations should be evaluated at the film temperature:

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

Correlations for different surface geometries

Vertical surfaces (Churchill and Chu correlation, valid for all $Ra \leq 10^{12}$):

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

This is one of the most commonly used correlations in natural convection. It works for any Prandtl number and covers both laminar and turbulent regimes in a single expression.

Horizontal surfaces, hot side facing up (unstable configuration):

• $\overline{Nu} = 0.54 \, Ra^{1/4}$ for $10^4 \leq Ra \leq 10^7$
• $\overline{Nu} = 0.15 \, Ra^{1/3}$ for $10^7 \leq Ra \leq 10^{11}$

The characteristic length for horizontal plates is $L = A_s / P$, where $A_s$ is the surface area and $P$ is the perimeter.

Horizontal surfaces, hot side facing down (stable configuration):

• $\overline{Nu} = 0.27 \, Ra^{1/4}$ for $10^5 \leq Ra \leq 10^{11}$

Notice the coefficient (0.27) is much smaller than the upward-facing case (0.54), reflecting the suppressed convection in the stable configuration.

Inclined surfaces can often be treated using vertical-surface correlations with gravity replaced by its component along the surface. For the Churchill and Chu correlation applied to an inclined plate, replace $Ra$ with $Ra \cdot \cos(\theta)$, where $\theta$ is the angle of inclination from vertical. This approach works well for inclination angles up to about 60° from vertical.

Applying these correlations: step-by-step

1. Identify the surface geometry (vertical, horizontal, inclined) and heating configuration.

2. Calculate the film temperature: $T_f = (T_s + T_\infty)/2$.

3. Look up fluid properties ($\beta$, $\nu$, $\alpha$, $k$, $Pr$) at $T_f$.

4. Determine the characteristic length $L$ (height for vertical plates, $A_s/P$ for horizontal plates).

5. Calculate $Ra = \frac{g \beta (T_s - T_\infty) L^3}{\nu \alpha}$.

6. Select the appropriate correlation based on geometry, orientation, and $Ra$ range.

7. Compute $\overline{Nu}$ from the correlation.

8. Find the heat transfer coefficient: $h = \overline{Nu} \cdot k / L$.

9. Calculate the heat transfer rate: $q = h \cdot A_s \cdot (T_s - T_\infty)$.