Natural Convection

About this note

A plain-language summary of the 4 publications listed under this topic on Academic Publications (Topic-wise). Full citations are under References.

Why this matters

Natural convection is the flow that heat makes for itself. Warm fluid near a hot wall becomes lighter and rises, cool fluid near a cold wall sinks, and a circulation sets up with no pump or fan. It cools buildings and electronic cabinets, ventilates grain silos, and, in a nuclear reactor after an accident, may be what keeps a bed of hot debris from overheating. Its strength is measured by the Rayleigh number, a dimensionless ratio of the buoyant push to the damping effects of viscosity and heat diffusion; the larger it is, the more vigorous the flow.

A recurring question in our work has been what happens when solid objects sit in the way of this flow: one block, many blocks, or a whole porous matrix. As the obstacles multiply, the enclosure starts to behave like a porous medium, and the papers here trace that path.

What we did

One block in a cavity

The starting point, with Prasad Bhave and D. A. S. Rees of the University of Bath, was a classic test case: a square cavity with one hot and one cold vertical wall and insulated top and bottom (Bhave et al., 2006). In such a cavity the centre holds a nearly stagnant core that does not take part in the circulation; worse, it conducts heat vertically from the warm fluid at the top to the cool fluid at the bottom, weakening the overall flow. We placed an adiabatic (perfectly insulating) block at the centre and, using finite volume simulations, varied its size, the Rayleigh number (10³ to 10⁶) and the Prandtl number (0.071, 0.71 and 7.1, spanning liquid metals, air and water).

Many blocks, and the porous view

With Jamalud-Din, Rees and B. V. K. Reddy, we filled the cavity with a uniform array of conducting square blocks and asked how much fluid circulates through the gaps between them (Jamalud-Din et al., 2010). We predicted the flow rate in two ways: a network model, which treats each gap as a small channel carrying a combination of pressure-driven and buoyancy-driven flow and links the channels together, and full numerical simulations. We also viewed the block-filled cavity as a porous medium, with porosity and permeability set by the number of blocks and the gap size.

A porous annulus that makes its own heat

With B. V. K. Reddy, we then studied a vertical annulus packed with a porous matrix that generates heat internally, while its side walls are held at different temperatures (Reddy and Narasimhan, 2010). This mimics, for example, a heat-generating debris bed cooled through a central downcomer. Because the solid and the fluid can be at different temperatures, we used a local thermal non-equilibrium model, which solves separate energy equations for the solid and the fluid, together with a general momentum equation that includes viscous, Darcy and form drag effects.

Pushing to high Rayleigh numbers

Conventional finite volume and finite element codes become prohibitively expensive for porous enclosures beyond a modified Rayleigh number of about 10⁴. With V. Babu, we used a parallel implementation of the lattice Boltzmann method (a computational approach that models fluid as particle distributions on a grid and parallelises naturally) to simulate a porous square enclosure at modified Rayleigh numbers up to 10⁸ and Darcy numbers (dimensionless permeability) from 10⁻⁶ to 10⁻² (Ramanathan et al., 2011).

What we found

A well-chosen block helps. Blocking the stagnant core stops its vertical heat conduction and increases the heat carried from the hot wall to the cold wall, but only up to a point. For each Rayleigh and Prandtl number there is an optimum block size, roughly equal to the conduction-dominated core of the empty cavity; a larger block starts to crowd out the convecting fluid. With a block occupying about half the cavity, heat transfer rose by 10% for a fluid of Prandtl number 0.071 at a Rayleigh number near 10⁶, with smaller gains elsewhere. Correlations for the optimum size and the maximum heat transfer matched the simulations within ±3% (Bhave et al., 2006).

A simple network model works, within limits. The network model agreed well with the simulations until the Rayleigh number multiplied by the cube of the dimensionless gap reached about 12. Viewed as a porous medium, adding more blocks at fixed porosity lowered the permeability and reduced the flow, while widening the gaps for a fixed number of blocks raised the porosity and the flow (Jamalud-Din et al., 2010).

Internal heating can split the flow in two. In the heat-generating annulus, the single circulation cell driven by the wall temperature difference changed into two cells once internal heating passed a critical value, because the hottest point moved from the hot wall into the interior. That critical value rose with the external Rayleigh number and with the solid-to-fluid conductivity ratio. When the solid conducts much better than the fluid, the simpler equilibrium model overpredicted heat transfer, by up to 61% in the case examined. A correlation for the annulus Nusselt number (a dimensionless heat transfer rate) predicted the results within ±20% (Reddy and Narasimhan, 2010).

Form drag matters at high Rayleigh numbers. The lattice Boltzmann results agreed with earlier experiments and simulations, to within ±0.5% in the clear fluid limit. At high Rayleigh numbers, including form drag (the resistance from flow around the solid matrix) gave Nusselt numbers up to 20% lower than the commonly used Darcy model, which is valid only for very low permeability and slow flows. We therefore defined a Rayleigh number modified by the form coefficient and proposed a power-law correlation for heat transfer that extends into the non-Darcy regime (Ramanathan et al., 2011).

Related

• Free Convection and the Rayleigh Number
• Free Convection for Dummies
• Essentials of Heat and Fluid Flow in Porous Media
• Bi-disperse Porous Media

References

1. Ramanathan, V., Narasimhan, A. and Babu, V., (2011), High Rayleigh Number Natural Convection Inside 2D Porous Enclosures Using the Lattice Boltzmann Method, ASME J. Heat Transfer, 133 (6), 062501 (9 pages), [DOI:10.1115/1.4003534]
2. Reddy, B. V. K. and Arunn Narasimhan, A., (2010), “Heat Generation Effects in Natural Convection inside a Porous Annulus”, Int. Comm. Heat and Mass Transfer, 37(6), 607-610. [DOI:10.1016/j.icheatmasstransfer.2009.12.018]
3. Sherifull-Din Jamalud-Din, Rees, D. A. S., Reddy, B. V. K. and Narasimhan, A., (2010), “Prediction of Natural Convection Flow Using Network Model and Numerical Simulations inside Enclosure with Distributed Solid Blocks”, Heat and Mass Transfer, 46(3), 333-343. [DOI 10.1007/s00231-009-0567-9]
4. Bhave, P., Narasimhan, A. and Rees, D. A. S., (2006), “Natural Convection Heat Transfer Enhancement using Adiabatic Block: Optimal Block Size and Prandtl Number Effect” Int. J. Heat and Mass Transfer, 49, 3807 – 3818. [DOI 10.1016/j.ijheatmasstransfer.2006.04.017]