Arunn Narasimhan

Arunn Narasimhan

Temperature Dependent Viscosity Flows and Convection

Created Last modified 9 min read

About this note

A plain-language summary of the 10 publications listed under this topic on Academic Publications (Topic-wise). Full citations are under References.

Why this matters

The viscosity of most liquids, their resistance to flow, depends strongly on temperature: warm honey or warm oil runs far more easily than cold. Yet analyses of forced convection through porous media almost always assumed a uniform viscosity. This research line, carried out with José Lage at the Laboratory for Porous Materials Applications, Southern Methodist University, Dallas, set out to see what happens when that assumption is dropped.

The motivation was practical. A cold plate for cooling airborne military phased-array radars used a porous insert of compressed aluminium foam, cooled by poly-alpha-olefin (PAO), a synthetic oil that is also the base-stock for motor oils and whose viscosity falls sharply as it warms. The aircraft’s environmental control unit could supply only a limited pressure drop across the cold plate, and the porous insert raised the pressure drop considerably, so designers needed to predict it accurately for a liquid that is heated as it flows (Narasimhan and Lage, 2004a). Similar questions arise in porous journal bearings, geothermal energy extraction and oil exploration (Narasimhan and Lage, 2001a; Narasimhan and Lage, 2002a).

The standard tool is the Hazen-Dupuit-Darcy (HDD) model, often called the Forchheimer-extended Darcy equation, a name Lage has shown to be historically incorrect. Darcy found that the flow through a sand layer is proportional to the pressure drop; Hazen showed that the fluid viscosity enters this relation; and Dupuit added a term growing with the square of the speed for form drag, the resistance that arises as fluid is forced around the solid obstacles. The model has two drag terms: a viscous drag, set by viscosity and permeability (a measure of how easily fluid passes through a porous material), and a form drag, set by density and a form coefficient. Crucially, all of this was developed from experiments in which the fluid was neither heated nor cooled (Narasimhan and Lage, 2001b).

What we did

Why the classic model fails

We simulated PAO flowing through a parallel-plate porous channel whose walls supply a uniform heat flux, and compared the computed pressure drop with HDD predictions using three reasonable choices of “representative” viscosity: at the average bulk temperature, at the log-mean temperature, and averaged along the channel. The best choice came within about 10%, but it gave the wrong shape for the pressure drop versus speed curve, so it would diverge further over a wider range (Narasimhan and Lage, 2001b).

The reason was surprising. Heating makes the viscosity vary across the channel, which distorts the velocity profile, and this alters not only the viscous drag but also the form drag, a term long believed to be independent of viscosity. We therefore proposed the Modified Hazen-Dupuit-Darcy (M-HDD) model, which multiplies each drag term by a correction coefficient and gives empirical relations for both as functions of heat flux. The model revealed a sequence of regimes as heating increases: first both drag terms are affected, then only the form drag, until finally the viscous effect vanishes in a “heating form-drag limit” (Narasimhan and Lage, 2001b).

Effect on heat transfer

The same simulations showed that although heating strongly reshapes the velocity profile, it barely changes the temperature profile, so the Nusselt number (a dimensionless measure of convective heat transfer) increases only very slightly compared with a uniform-viscosity fluid (Narasimhan and Lage, 2001a). This contrasts with earlier flat-plate studies that found a strong effect, a difference we traced to their omission of form drag. Without form drag the predicted pressure drop would fall without bound as the fluid heats up. The departure from uniform-viscosity behaviour also occurs over only a short length of channel, a length inversely proportional to the wall heat flux. We introduced a Nusselt number based on the surface-averaged wall temperature and the inlet fluid temperature, which are easier to measure than the quantities in the usual definition.

A theory, and an experiment to test it

With Donald Nield (University of Auckland), we developed a perturbation theory, a systematic series of zero-, first- and second-order approximations, for the velocity, temperature, pressure drop and Nusselt number in fully developed flow (Narasimhan et al., 2001b). Using the HDD model with viscosity taken at the maximum or minimum fluid temperature failed drastically, while the new theory matched numerical results, its Nusselt number predictions being very accurate when the ratio of form to viscous drag was below 0.3.

Both models were then tested experimentally, with Dan Porneala, on a cold plate with a compressed aluminium-alloy foam insert, heated electrically and cooled by PAO (Narasimhan et al., 2001a). The foam’s low permeability and high form coefficient made it a demanding test. The second-order theory improved on an earlier Darcy-only model but was accurate only when the temperature varied little along the channel, deviating systematically at higher heat flux. The M-HDD model agreed extremely well with the measurements at all heat fluxes tested. We presented both models together, with their numerical and experimental validation, in an American Mathematical Society volume (Narasimhan and Lage, 2002a), and a further paper in the series examined the thermo-hydraulics of heated porous channel flows (Narasimhan and Lage, 2003).

Inlet temperature and the departure from Darcy flow

At low speeds the pressure drop rises linearly with speed (Darcy flow); at higher speeds form drag dominates and the relation becomes quadratic. For uniform viscosity the transition is usually placed where the two drag terms are equal. Using the M-HDD model, we showed that heating moves this transition to lower speeds, ever lower as the heat flux rises, until the flow is practically always form-drag dominant (Narasimhan and Lage, 2002b). The inlet temperature matters: for one heat flux and speed, PAO entering at 7°C was already form-drag dominant, while at 21°C and 32°C it was still in the linear regime. Fresh simulations at these three inlet temperatures led to generalised correlations for the M-HDD coefficients that include inlet temperature, extending the model to other liquids whose viscosity follows a similar power law with temperature (Narasimhan and Lage, 2004b).

Can heating save pumping power?

Heating a liquid lowers its viscosity and therefore the pumping power, but heat costs energy too. We defined a figure of merit, the pump power saved divided by the heat power spent, and evaluated it with the M-HDD model (Narasimhan and Lage, 2004a). Colder inlet liquid gave a more efficient process, and raising the heat flux brought diminishing returns because the liquid becomes hot soon after entering the channel. The porous channel reached about 30% overall energy efficiency at the lowest heat flux and inlet temperature, falling below 3% at the highest inlet temperature tested (32°C).

What we found

Taken together, and reviewed in a chapter of the Handbook of Porous Media (Narasimhan and Lage, 2005), the work leads to a few clear lessons:

  • When a liquid’s viscosity changes with temperature, heating alters both the viscous and the form drag in a porous medium, so the classic HDD model cannot predict the pressure drop, whichever single viscosity is chosen.
  • The M-HDD model captures this and is believed to be valid for any porous medium configuration, although its coefficient correlations were derived for PAO and apply to liquids with similar viscosity behaviour.
  • Velocity is strongly affected while heat transfer is only slightly affected, so form drag must always be included.
  • Heating moves the departure from Darcy flow to lower speeds, less so for hotter inlet liquid, and heating can be an effective way to reduce pumping power in porous medium flows.

The chapter also notes what remains open: the analyses assume a rigid porous medium and constant liquid density, and the correlations should be tested in other geometries, media and liquids to extend their use to devices such as cold plates and self-lubricating bearings (Narasimhan and Lage, 2005).

References

  1. Narasimhan, A. and Lage, J. L., (2005), “Variable Viscosity Forced Convection in Porous Medium Channels” in Handbook of Porous Media 2nd edition (Ed. K. Vafai), 5, 195-233, Taylor and Francis pub., NY.
  2. Narasimhan, A. and Lage, J. L., (2004a), “Pump Power Gain for Heated Porous Medium Channel Flows”, ASME J. Fluids Engineering, 126, 494-497. [DOI 10.1115/1.1760546]
  3. Narasimhan, A. and Lage, J. L., (2004b), “Predicting Inlet Temperature Effects on the Pressure-drop of Heated Porous Medium Channel Flows Using the M-HDD Model”, ASME J. Heat Transfer, 126, 301-303. [DOI 10.1115/1.1667526]
  4. Narasimhan, A. and Lage, J. L., (2003), “Temperature-Dependent Viscosity Effects on the Thermo-hydraulics of Heated Porous Medium Channel Flows”, J. Porous Media, 6, 149-158. [DOI 10.1615/JPorMedia.v6.i3.10 ]
  5. Narasimhan, A. and Lage, J. L., (2002a), “New Models for Predicting Temperature-Dependent Viscous Effects on Flow through Porous Media” in Fluid Flow and Transport in Porous Media: Mathematical and Numerical Treatment, (Eds. Z. Chen and R. Ewing), American Mathematical Society Contemporary Mathematics (CONM) Book Series, 397-408.
  6. Narasimhan, A. and Lage, J. L., (2002b), “Inlet Temperature Influence on the Departure from Darcy Flow by Fluids with Variable Viscosity”, Int. J. Heat and Mass Transfer, 45, 2419-2422. [DOI 10.1016/S0017-9310(01)00324-6]
  7. Narasimhan, A. and Lage, J. L., (2001a), “Forced Convection of a Fluid with Temperature-Dependent Viscosity through a Porous Medium Channel”, Numerical Heat Transfer – Part A, 40(8), 801-820. [DOI 10.1080/104077801753344277]
  8. Narasimhan, A., Lage, J. L., Nield, D. A. and Porneala, D. C., (2001a), “Experimental Verification of Two New Theories for Predicting the Temperature-Dependent Viscosity Effects on the Forced Convection through a Porous Media Channel”, ASME J. Fluids Engineering, 123, 948-951. [DOI 10.1115/1.1413245]
  9. Narasimhan, A., Lage, J. L., and Nield, D. A., (2001b), “New Theory for Forced Convection through Porous Media by Fluids with Temperature-Dependent Viscosity”, ASME J. Heat Transfer, 123, 1045-1051. [DOI 10.1115/1.1409268]
  10. Narasimhan, A. and Lage, J. L., (2001b), “Modified Hazen-Dupuit-Darcy Model for Forced Convection of a Fluid with Temperature-Dependent Viscosity”, ASME J. Heat Transfer, 123, 31-38. [DOI 10.1115/1.1332778]

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