Draft:Scale analysis of natural convection in tall enclosures

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Scale analysis of natural convection in tall enclosures

Natural convection is a heat transfer process driven by buoyancy forces, occurring when fluid motion results from temperature-induced density variations. In tall enclosures, where the height is significantly greater than the width, natural convection exhibits distinct flow behaviors due to the geometric constraints. These enclosures are commonly found in applications such as solar chimneys, heat exchangers, and building ventilation systems. Understanding the dynamics in these environments is crucial for optimizing thermal performance and energy efficiency. A scale analysis simplifies complex governing equations and helps predict heat transfer and fluid flow patterns in tall enclosures by focusing on dimensionless parameters like the Rayleigh and Nusselt numbers. This approach aids in designing efficient systems for various industrial and engineering applications.

Tall Enclosures

A tall enclosure refers to a cavity where the height (H) is much larger than the width (L), typically H/L ≥ 5. Examples include vertical channels, chimneys, and tall storage tanks. The high aspect ratio (H/L) leads to distinct natural convection patterns. In such enclosures, the boundary layers near the vertical walls are the primary regions where convection occurs.

In tall enclosures:

Vertical Boundary Layers: The temperature difference between the walls and the fluid drives buoyant flow. It rises as the fluid heats up near the warmer wall, while the more astounding wall causes descending fluid motion.

Core Region: In the middle of the enclosure, far from the walls, fluid flow may be relatively stagnant or form weak recirculation zones.

Aspect Ratio Effects: A high aspect ratio means the boundary layers dominate the flow, while the core region becomes more passive regarding heat transfer.

The scale analysis of natural convection in tall enclosures involves studying how critical parameters like temperature, velocity, and length scales behave in relation to each other. Tall enclosures, such as buildings, chimneys, or even geophysical formations, are common in engineering and environmental contexts.

1. Governing Equations

The primary equations governing natural convection are the Navier-Stokes equations (for momentum) and the heat transport equation (for energy).

Continuity equations

${\frac {\partial u}{\partial x}}+{\frac {\partial u}{\partial y}}=0$

Momentum Equation (Boussinesq Approximation)

Using the Boussinesq approximation, the momentum equation can be written as:

Vertical momentum:

$\rho \left(u{\frac {\partial u}{\partial x}}+v{\frac {\partial u}{\partial y}}\right)=-{\frac {\partial P}{\partial x}}+\mu \nabla ^{2}u-\rho g$

Horizontal momentum:

$\rho \left(u{\frac {\partial v}{\partial x}}+v{\frac {\partial v}{\partial y}}\right)=-{\frac {\partial P}{\partial y}}+\mu \nabla ^{2}v$

Energy Equation(Thermal Transport):

$u\cdot {\frac {\partial T}{\partial x}}+v\cdot {\frac {\partial T}{\partial y}}=\alpha \Delta ^{2}\mathrm {T}$

where, T=temperature

$\alpha$=Thermal diffusivity

2. Characteristic Scales

To perform scale analysis, characteristic scales are introduced:

• Vertical Length Scale: H
• Horizontal Length Scale: L
• Temperature Difference: ΔT (between hot and cold walls)
• Velocity Scale: U
• Pressure Scale: Δp
• The ratio of height to width defines the aspect ratio: A = H / L In a tall enclosure, A >> 1.

3. Dimensionless Groups

Important dimensionless groups arise from scaling the governing equations:

Rayleigh Number (Ra):

$Ra_{H}={\frac {g\cdot \beta \cdot \Delta T\cdot H^{3}}{\alpha \cdot \nu }}$

• Where:
  • g = gravitational acceleration
  • β = thermal expansion coefficient
  • ν = kinematic viscosity

Prandtl Number (Pr):

$P_{r}={\frac {\upsilon }{\alpha }}$

For natural convection, Ra and Pr dictate the flow regime (laminar or turbulent) and temperature distribution.

4. Scaling Analysis

Vertical Momentum Balance:

By balancing the buoyancy term with the inertial or viscous term, we obtain:

Inertial vs Buoyancy:

$\rho \cdot {\frac {U^{2}}{H}}=\rho \cdot g\cdot \beta \cdot \Delta T$

This gives the velocity scale as

$U\approx {\sqrt {g\cdot \beta \cdot \Delta T\cdot H}}$

Viscous vs Buoyancy:

$\mu \cdot \left({\frac {U}{\delta ^{2}}}\right)\approx \rho \cdot g\cdot \beta \cdot \Delta T$

Here, solving for $\delta$, boundary layer thickness,

$\delta \approx \left({\frac {\nu \cdot H}{\sqrt {g\cdot \beta \cdot \Delta T\cdot H}}}\right)^{\frac {1}{2}}$

Horizontal momentum balance:

Since the horizontal dimensions are small compared to the vertical, the flow is dominated by viscous forces in this direction.

Energy Equation:

By balancing convection and diffusion in the energy equation, the thermal boundary layer thickness $\delta \tau$ can be estimated. For tall enclosures:

$\delta _{T}\approx H\cdot Ra^{-1/4}$

5. Flow Regimes

Laminar Convection:

The flow remains laminar with smooth temperature gradients for low Rayleigh numbers (typically $Ra<10^{9}$).

Turbulent Convection:

At higher Rayleigh numbers, the flow becomes turbulent, leading to enhanced heat transfer and mixing.

6. Heat Transfer Scaling

The heat transfer is often characterized by the Nusselt Number (Nu), which relates convective to conductive heat transfer:

For tall enclosures:

Laminar regime:

$Nu\approx Ra^{\frac {1}{4}}$

Turbulent regime:

$Nu\approx Ra^{\frac {1}{3}}$

Conclusion

The scale analysis of natural convection in tall enclosures shows that the Rayleigh number primarily dictates flow behavior, and the heat transfer efficiency can be characterized by the Nusselt number. In tall enclosures, vertical buoyancy forces dominate the flow, with boundary layers forming near the walls. The height of the enclosure leads to significant vertical stratification, while horizontal variations are more subdued due to the small width.

References

1.Scales of natural convection on a convectively heated vertical wall | Physics of Fluids | AIP Publishing

2.https://www.cambridge.org/core/journals/journal-of-fluid-mechanics/article/geophysicalscale-model-of-vertical-natural-convection-boundary-layers/C42C5CD82A0200282DA8A4FA18ACFAD6

3.https://www.sciencedirect.com/science/article/pii/S1738573316300080

Article prepared by

1. Isha Dhamdhere, Roll no.: 21135050, IIT (BHU) Varanasi
2. Manisha Bishnoi, Roll no.: 21134016, IIT (BHU) Varanasi
3. Anushka Singh Jaiswal, Roll no.: 21135024, IIT (BHU) Varanasi
4. Misha Gupta, Roll no.: 21135081, IIT (BHU) Varanasi
5. Raksha Nehra, Roll no.: 21135110, IIT (BHU) Varanasi