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Nanofluid Thermal Conductivity and Mixing Rules
Maxwell’s mixing rules refute the notion that nanofluids comprising nanoparticles in common solvents enhance heat transfer by increasing thermal conductivity
Over the past decade, nanofluids have been of great interest because experiments have shown small quantities of nanoparticles (NPs) significantly increase the thermal conductivity of common solvents. However, some tests show nanofluids do not enhance thermal conductivity while others surprisingly show the thermal conductivity to be less than that of the solvent. Moreover, nanofluids do not obey Maxwell mixing rules where the thermal conductivity is given by the weighted average of the NP and solvent conductivities. Why the thermal conductivity of nanofluids is anomalous and violates Maxwell’s mixing rules are unresolved problems in condensed matter physics.
Regardless, nanofluids should obey Maxwell’s mixing rules. Therefore, the increased heat transfer of nanofluids is not caused by an increase in thermal conductivity, but rather is just what it appears to be – an increase in heat transfer. For example, stirring a liquid increases its heat transfer, but not its thermal conductivity. Similarity suggests increased heat transfer in nanofluids has nothing to do with thermal conductivity. If so,
What is the mechanism by which NPs increase heat transfer?
Heat Transfer Mechanisms
Brownian motion of the solvent around randomly moving NPs is a form of heat transfer, but being limited to nanoscale distances cannot significantly enhance heat transfer. Convection moves heat over larger distances, but is not relevant in stationary nanofluids. However, electromagnetic (EM) radiation differs. By Beers law, EM radiation allows heat to be transferred over large distances depending on the absorptivity of the solvent, but
How is EM radiation produced by NPs?
QED Induced Radiation
At ambient temperature, the EM energy available in nanofluids is available at the IR peak near 10 microns. Inelastic collisions of solvent molecules therefore transfer IR energy to the NPs. Classically the NP conserves the absorbed IR by an increase in temperature that in turn is re-emitted as IR radiation. Since most solvents are highly absorptive in the IR, the re-emitted IR is absorbed by the solvent very close to the NP surface. Lacking penetration, heat trransfer is not enhanced and the NP may be said to be in local thermal equilibrium with the solvent.
Quantum mechanics (QM) differs.
QM by the Einstein-Hopf relation requires NPs have vanishing heat capacity, and therefore NPs cannot conserve the absorbed IR energy by an increase in temperature. Because of the high surface to volume ratio of NPs, however, the absorbed IR concentrates in the NP surface. Therefore, the NP is placed under TIR confinement, the consequence of which allows QED to induce the creation of low intensity EM radiation inside the NP. TIR stands for total internal reflection and QED for quantum electrodynamics.
In NPs, QED creates short wavelength EM radiation, the wavelength given by 2nd, where n and d are the refractive index and diameter of the NP. By their submicron size, the NPs therefore create EM radiation from the VUV to the VIS that leaks into the solvent. Depending on the absorption spectrum of the solvent, the EM radiation may significantly penetrate the solvent, and if so, the NPs enhance heat transfer by transferring heat over a greater distance than if the NPs were not there as in the solvent alone. In effect, heat transfer is enhanced because NPs violate local thermal equilibrium. See http://www.nanoqed.org at “Nanofluid Thermal Conductivity,”
Experimental Methods Nanofluid experiments are usually performed with thermal conductivity measured using  the transient hot wire (THW) or  the steady state parallel plate (GAP) method. However, the tests are designed to measure thermal conductivity - not heat transfer. Both THW and GAP methods impose a known heat source and measure the temperature difference over small distances, and as such require interpretation to infer heat transfer enhancements. Perhaps, a better method  is to impose a temperature difference across a gap filled with the nanofluid and measure the heat transfer directly with a heat flux meter.
Gold NPs < 50 nm Recent THW and GAP tests  on nanofluids of 2-45 nm gold NPs in a methanol/water solvent showed virtually no increase in thermal conductivity over the solvent. Moreover, the 2 nm NPs showed a decrease in conductivity. However, gold has a low refractive index n = 0.17 compared to other materials having n > 1. Therefore, QED induces EM radiation in the VUV at wavelengths from 1-15 nm while water in this range has a high absorption at VUV wavelengths < 20 nm. Since penetration of EM radiation is inversely proportional to absorption, the 2-45 nm gold NPs consistent with  provide little, if any heat transfer enhancement. However, the reported decrease in thermal conductivity for 2 nm NPs is suspect as Maxwell’s mixing rules preclude the conductivity of the nanofluid to be less than that of water.
The negligible enhancement  for 2-45 nm gold NPs should not be taken as a general conclusion for other materials. For example, silica NPs having n = 1.44 are expected to create EM radiation in the UV near the VIS window where water is transparent, and therefore silica should show far greater enhancement than for gold.
Thermocouple Placement In the THW and GAP tests, the separation between the heat source and thermocouple (TC) is 5-15 mm and 200-300 microns, respectively. QED induced EM radiation from the 2-45 nm gold NPs  that is absorbed in the solvent before reaching the TC provides valid estimates of heat transfer. However, for silica NPs > 50 nm, the heat transfer should be understated as the EM radiation is not absorbed before reaching the TC.
1. Thermal conductivity of nanofluids is required to obey Maxwell mixing rules.
2. THW and GAP tests to determine nanofluid thermal conductivity need not be performed as conductivity may be determined by Maxwell’s mixing rules without any loss of accuracy
3. Heat transfer and not thermal conductivity is the proper measure of nanofluids and may be determined by measuring the heat flux upon imposing temperatures across the nanofluid placed between parallel flat plates.
4. THW and GAP tests having small separations between the heat source and the TCs are valid for 2-45 nm gold NPs but most likely understate the enhancement of heat transfer in NPs > 50 nm of materials other than gold.
5. Enhancement of heat transfer in nanofluids depends on the refractive index and diameter of the NPs in relation to the absorption spectrum of the solvent.
6. QM that requires the heat capacity of NPs to vanish precludes conservation of inelastic solvent molecule collisions by an increase in temperature. Instead, conservation proceeds by QED inducing the NPs under TIR confinement to create EM radiation that is absorbed at a penetration depending on the absorption spectrum of the solvent.
7. NPs enhance heat transfer of solvents by transferring the thermal energy of inelastic molecular collisions over the penetration depth of QED induced EM radiation into the solvent.
 J. Eapen, et al., Mean-Field Versus Microconvection Effects in Nanofluid Thermal Conduction, PRL 99, 095901 (2007)
 N. Shalkevich, et al, On the Thermal Conductivity of Gold Nanoparticle Colloids, Langmuir, 26, 663–670 (2010)
 A. Narayanaswamy, et al., Breakdown of the Planck blackbody radiation law at nanoscale gaps, Appl Phys A, 96, 357–362 (2009)
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Classically, thermal EM radiation conserves heat by an increase in temperature. But at the nanoscale, temperature increases are forbidden by quantum mechanics. QED radiation explains how heat is conserved by the emission of nonthermal EM radiation.
Page Updated Last on: Apr 12, 2012