Arunn Narasimhan

Arunn Narasimhan

Bi-disperse Porous Media

Created Last modified 6 min read

About this note

A plain-language summary of the 5 publications listed under this topic on Academic Publications (Topic-wise). Full citations are under References.

Why this matters

A porous medium is a solid riddled with connected holes through which a fluid can move: a sponge, a bed of sand, a stack of heat sink fins. Most of the theory treats the pores as having a single typical size. Many real materials, however, have two. Take clusters of large grains that are themselves made of smaller grains, or a coal pile made of porous coal particles; cooked rice, with water soaked into each grain and larger gaps between the grains, is a kitchen example. Such a material is called a bi-disperse porous medium (BDPM): a porous medium with large pores (macropores) whose solid part is itself a porous medium with much smaller pores (micropores), the same fluid filling both (Narasimhan, 2019).

BDPMs appear in aquifers, catalyst pellets, fuel cells, heat pipe wicks and microreactors. We were drawn to them by a different idea: a whole assembly of heat-producing parts, such as electronic components in a cabinet or servers in a data centre, can itself be viewed as a BDPM. If that view holds, the well-developed tools of porous-medium heat transfer can give quick design answers without resolving every component in a full computational fluid dynamics model.

What we did

With B. V. K. Reddy, we built a simple, controllable model of a BDPM: a square box or channel filled with a regular array of square porous blocks separated by gaps. The gaps play the role of the macropores and the blocks supply the micropores. The strength of the bi-dispersion could then be tuned with a few parameters: the permeability of the blocks (permeability being a measure of how easily fluid passes through a porous material, expressed here as an internal Darcy number), the fraction of the volume taken by the gaps, and the number of blocks. We solved the equations for mass, momentum and energy numerically in both the gaps and, in volume-averaged form, inside the blocks.

The two limits frame everything. When the blocks become impermeable, the box is an ordinary single-scale (mono-disperse) porous medium in which fluid moves only through the gaps. When the blocks become fully permeable, they vanish and the box holds only clear fluid. A BDPM sits between these extremes.

We first applied this to natural convection, where flow is driven only by buoyancy, in a square enclosure heated on one side wall and cooled on the other (Narasimhan and Reddy, 2010). Next we made the heating pulsate in time to look for resonance, the condition in which the heating frequency matches the natural rhythm of the convection flow and heat transfer peaks (Narasimhan and Reddy, 2011a). We then turned to forced convection, where a pump drives the flow, in a channel whose porous blocks generate heat, standing in for electronics being cooled (Narasimhan and Reddy, 2011b). With Pradip Dutta of the Indian Institute of Science, we extended this to a configuration resembling a data centre, in which the room is the macro-porous medium and each server, with its internal gaps, is a micro-porous block (Narasimhan et al., 2011).

Finally, for a CRC Press volume on convective heat transfer in porous media, we reviewed the research on convection in BDPMs published mostly after 2010, including forced, natural and mixed convection, phase change and the newer tri-disperse media with three levels of porosity (Narasimhan, 2019).

What we found

Natural convection. Letting fluid pass through the blocks raises the heat transfer across the enclosure, measured by the Nusselt number (the ratio of the actual heat transfer to what conduction alone would carry). The effect of bi-dispersion is strongest at low Rayleigh numbers (weak buoyancy) and fades as buoyancy grows; at high Rayleigh numbers the blocks in the core of the box hardly matter. Increasing the number of blocks at a fixed gap fraction lowers the permeability of the enclosure and reduces the heat transfer. We proposed a correlation for the Nusselt number in terms of a modified Rayleigh number and the enclosure permeability that predicts the simulations within 15% and 9% over two successive ranges (Narasimhan and Reddy, 2010).

Resonance. Natural convection resonance does occur in a BDPM enclosure. The resonance frequency increases with the Rayleigh number and with both the internal and external permeability, but always stays below that of the corresponding clear-fluid enclosure. A scale analysis, adapted to include bi-dispersion, predicted the resonance frequency in good agreement with the simulations (Narasimhan and Reddy, 2011a).

Cooling heat-generating blocks. For a fixed flow rate, raising the block permeability from to (as an internal Darcy number) increased the Nusselt number about four times, from about 200 to about 800, while the pressure drop fell almost eightfold. Below the benefit collapsed toward the mono-disperse limit. Many small, well-distributed blocks did better than one large block, with very little extra pumping power (Narasimhan and Reddy, 2011b).

Data centre analogue. When the gap fraction was raised from 0.2 to 0.86, heat transfer doubled (Nusselt number from about 550 to about 1100) and the pressure drop again fell almost eightfold. Beyond about 49 blocks, further subdivision brought little further gain. The pumping power stayed almost unchanged across the range of block numbers studied (Narasimhan et al., 2011).

The wider picture. The review concludes that for both pore scales to matter, their sizes should differ by between one and three orders of magnitude; when they differ by much more, the larger pores dominate and the medium behaves like a mono-disperse one. It also identifies open questions, including thermal non-equilibrium between phases, interfaces between pore scales, phase change, and above all the need for experiments, without which much BDPM convection theory will remain impractical for engineering (Narasimhan, 2019).

References

  1. Narasimhan, A., (2019), “Convection in Bi-Disperse Porus Media,” Ch. 8, in Convective Heat Transfer in Porous Media, eds. Yasser Mahmoudi, Kamel Hooman, Kambiz Vafai, CRC Press, Taylor & Francis Group. ISBN 9780367030803. [ Web link ]
  2. Narasimhan, A., Reddy, B. V. K. and Dutta, P. (2011), “Thermal Management using the Bi-disperse Porous Medium Approach,”  Int. J. of Heat and Mass Transfer, 55(4), 538–546. [DOI 10.1016/j.ijheatmasstransfer.2011.11.006]
  3. Narasimhan, A. and Reddy, B. V. K., (2011a), “Resonance of Natural Convection inside a Bi-disperse Porous Medium Enclosure”, ASME J. Heat Transfer, 133 (4), 042601 (9 pages). [DOI: 10.1115/1.4001316]
  4. Narasimhan, A. and Reddy, B. V. K., (2011b), Laminar Forced Convection in a Heat Generating Bi-Disperse Porous Medium Channel, Int. J. Heat and Mass Transfer, 54 (1-3), 636-644. [DOI: 10.1016/j.ijheatmasstransfer.2010.08.022]
  5. Narasimhan, A. and Reddy, B. V. K., (2010), “Natural Convection inside a Bi-disperse Porous Medium Enclosure”, ASME J. Heat Transfer, 132, 012502, (9 pages). [DOI:10.1115/1.3192134]

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