Electrical Conductivity Measurements of Nanofluids

Copyright © 2011 American Scientific Publishers All rights reserved Printed in the United States of America

Journal of Nanoscience and Nanotechnology Vol. 11, 1–8, 2011

Electrical Conductivity Measurements of Nanofluids and Development of New Correlations Hanumantharao Konakanchi, Ravikanth Vajjha, Debasmita Misra, and Debendra Das∗ College of Engineering and Mines, University of Alaska Fairbanks, P.O. Box 755905, Fairbanks, AK, 99775-5905, USA In this study the electrical conductivity of aluminum oxide (Al2 O3 , silicon dioxide (SiO2 and zinc oxide (ZnO) nanoparticles dispersed in propylene glycol and water mixture were measured in the temperature range of 0 C to 90 C. The volumetric concentration of nanoparticles in these fluids ranged from 0 to 10% for different nanofluids. The particle sizes considered were from 20 nm to 70 nm. The electrical conductivity measuring apparatus and the measurement procedure were validated by measuring the electrical conductivity of a calibration fluid, whose properties are known accurately. The measured electrical conductivity values agreed within ±1% with the published data reported by the manufacturer. Following the validation, the electrical conductivities of different nanofluids were measured. The measurements showed that electrical conductivity of nanofluids increased with an increase in temperature and also with an increase in particle volumetric concentration. For the same nanofluid at a fixed volumetric concentration, the electrical conductivity was found to be higher for smaller particle sizes. From the experimental data, empirical models were developed for three nanofluids to express the electrical conductivity as functions of temperature, volumetric concentration and the size of the nanoparticles.

Keywords: Electrical Conductivity, Nanofluids, Nanoparticles, Particle Size, Propylene Glycol,

1. INTRODUCTION In the past decade there has been a substantial amount of research on nanofluids, which are dispersions of nanoscale particles in regular fluids. Extensive experimental and theoretical studies including those of Jwo et al.,1 Yu et al.,2 Vajjha et al.3 have established that nanofluids have substantially higher thermal conductivity than their basefluids, in which the particles are dispersed. An impact of this phenomenon is reflected in enhancing the convective heat transfer coefficient of nanofluids. Therefore, nanofluids can be used as a successful coolant in many industrial applications, such as the plasma arc cutters (PAC), which operate at very high temperature of the order of 25000 K (Ramakrishna and Rogozinski).4 For many materials their thermal and electrical conductivities behave in similar manners. Therefore, it is expected that just as the thermal conductivity of nanofluid is higher than their basefluid, similarly the electrical conductivity of nanofluids may be higher than those of their basefluids. In plasma arc cutters, the electrodes generate the plasma arc and are subjected to intense heat and must require ∗

Author to whom correspondence should be addressed.

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direct cooling. Nemchinsky and Severance5 describe that an important factor influencing the erosion of cathode in plasma arc cutting is the cooling of the electrode. The better it is cooled the lower is the erosion rate. On the higher power side, there are mechanized cutting designed to use high currents of the order of 400 A for oxygen cutting and about 1000 A for argon–hydrogen mixture cutting. Water is commonly used as coolant but for operation at low temperatures, ethylene glycol or propylene glycol and water mixtures are necessary. The electrodes are hollow milled and a liquid tube is located in the center to provide a high flow velocity of the coolant in the passage. The coolant convects away the high degree of heat from the interior of the electrode and torch parts. In this type of torch design, the nozzle which is the outer annular structure surrounding the electrode is also cooled by liquid. As the demand on thickness, speed, strength and precision of metal cutting advances, the heat flux generated within the metal body of PAC grows in intensity. Therefore, better heat transfer fluids are necessary to prevent possible overheating. An overheated plasma arc cutting system will wear out the torch parts prematurely and may burn out the torch. Therefore, nanofluids may be better coolants than their basefluid for

1533-4880/2011/11/001/008

doi:10.1166/jnn.2011.4217

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Volumetric Concentration.

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Electrical Conductivity Measurements of Nanofluids and Development of New Correlations

this application. They will be able to maintain the temperatures of the cathode and nozzle at the desired level at a lower flow rate thus needing smaller passages and lower pumping power. On the other hand, the metal cathode and nozzle can be cooled to a lower temperature by nanofluids at the same volumetric flow as conventional coolant, thus diminishing the erosion rate. Ramakrishna and Rogozinski4 explain that for most efficient performance of plasma arc cutters, the electric power (V × I) of the arc (V is the voltage of the plasma at the nozzle exit with respect to cathode and I is the arc current) should go into gas heating within the torch, without being lost to the nozzle wall. Therefore, there is a possibility that the higher electrical conductivity of nanofluids may affect this electrical power. Since the fluid flows in the passages in metallic cathode and nozzle, which serves as the anode, the electrical conductivity of the liquid may influence the life span of electrodes. Therefore, it is essential that the designers know the electrical conductivity values of nanofluids accurately. With this knowledge, trade-off studies can be conducted by comparing the advantage gained by more efficient cooling of electrodes versus the effect of higher electrical conductivity on erosion of electrodes. To fulfill this need, good data on electrical conductivity of nanofluids as a function of nanoparticle material, average particle size, volumetric concentration and temperature are necessary. However, such information on electrical conductivity of nanofluids is not available in the current literature. Therefore, the objective of this present study was to measure the electrical conductivity of several nanofluids and develop suitable correlations, which can be used in applications such as the PAC. Additionally, the data from this study will fulfill the need in other applications, where electrical conductivity of fluids plays a vital role. Examples are: liquid flow in microchips containing semiconductors, electronic cooling systems, colloidal science, paints, polishing process and heat exchangers in power plants and heavy industries. Cruz et al.6 have shown that the electrical conductivity of fluids containing average particle size of 500 nm and additives of varying ionic concentrations, strongly affect the stability of suspensions. Since a stable dispersion is very important for nanofluids, knowing its electrical conductivity accurately can be beneficial in designing stably suspended fluid suspensions. Propylene glycol and water mixture (PG/W 60:40 by mass) is widely used as a heat transfer fluid in cold regions, such as Alaska, where use of water will lead to failure and damage of systems due to freezing. Therefore, our measurements have focused on propylene glycol based nanofluids.

2. THEORETICAL INFORMATION Wang and Hirata7 presented an equation for the electrical conductivity of colloidal suspensions. The equation is a 2

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summation of two conductivities due to ionic motion and particle motion and was expressed as nf = bf 1 − + Ze

3 4r 3

(1)

where Z is the valence of a charged particle, e is the charge on an electron, is the electrophoretic mobility of charged particles and r is the radius of particle. They made measurements using -alumina powder of median particle size of 200 nm. They used water as the base fluid with additions of 1N-HCl or 1N-NH4 OH to vary the pH of nanofluids between 2.0 to 10.0. Their measurements showed that the electrical conductivity was a function of the pH of the base fluid and nanofluid. The values of electrical conductivity of nanofluid nf were higher than those of the basefluid bf , due to the contribution of charged mobile alumina particles. Both values of nf and bf showed a minimum at pH in the range of 7 to 8. The electrophoretic mobility and the valence of charged alumina particle Z varied with pH. There were no measurements presented on the effect of the temperature in their study, therefore, their measurements were possibly taken at room temperature. Hayashi8 verified the electrical conductivity– temperature relation for natural waters proposed by Sorensen and Glass,9 which was linear in a temperature range of 0 to 30 C. t = 25 1 + at − 25

(2)

In Eq. (2) the temperature t is in C and 25 C was taken as a reference temperature. The electrical conductivities 25 and t are in S/cm at reference temperature and at t C respectively. The temperature correction factor a ( C−1 adopted by various researchers varied from 0.0191 to 0.025. For temperatures higher than 30 C, Hayashi recommended that a viscosity based equation of Sorensen and Glass was more accurate, because it accounted for the nonlinear viscosity-temperature relationship. t −b t = 25 (3) 25 In Eq. (3) t and 25 are the viscosity of pure water at temperatures t C and 25 C. Hayashi evaluated the exponent b from his measurements. It ranged between 0.806 and 0.933 with an average value of 0.877 based on the results of five different types of water having vastly different salinity and chemical composition. Note that the (2) and (3) equations are only valid for single phase liquid with no suspensions. Cruz et al.6 studied the electrical conductivity of aqueous suspensions of -alumina particles of average size 520 nm up to a 35% volumetric concentration. They varied the ionic strength of the nanofluids by adding appropriate amount of NH4 Cl. Their measurements showed that the electrical conductivity increased substantially with an J. Nanosci. Nanotechnol. 11, 1–8, 2011

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increase in the NH4 Cl additions. For the nanofluid without any NH4 Cl addition the conductivity increased with the increase in volumetric concentration of particles. By classifying the particles under three categories, p bf , p = bf , p bf they examined the interaction potential between a pair of particles. They presented the conditions of ionic strength that would make the suspensions very stable, and the ionic conditions that would make the suspension move toward coagulation. Their results did not show any measurements of electrical conductivity of suspensions at different temperatures. Cruz et al.6 also presented Maxwell’s model for the electrical conductivity of suspensions for spherical particles, which is given as Eq. (4). 2bf + p − 2bf − p nf = bf 2bf + p + bf − p

(4)

This model is based upon the assumption that the particles are randomly distributed and at distances much larger than their size. Therefore, there is no interaction among them and the electrical field surrounding each particle is undisturbed. Abedian and Baker10 have presented a correlation for electrical conductivity of single phase liquid by measuring the electrical conductivity and viscosity of five liquids used as gas turbine lubricants. The equation they presented is =

2ZCo R

(6)

In Eq. (5) Z, Co and R are ionic valence, concentration of charged species, a constant equal to 4 for polymer solution and mean molecular size respectively. The constants A and B in Eq. (6) are obtained from viscosity measurements. Glover et al.11 measured the electrical conductivity of single wall carbon nanotubes dispersed in 50% water and 50% ethylene glycol mixture. They observed a linear trend in the increase of the electrical conductivity with increase in weight percentage of carbon nanotubes. A 0.5 weight percent nanofluid exhibited 13 times the electrical conductivity of the base fluid. No measurements with temperature variation were noted. Wong and Kurma12 pointed out that presently there was insufficient data published on electrical conductivity of nanofluids and good data were needed for an accurate determination of this property. To initiate research on this property, they performed measurements of electrical conductivity of alumina nanofluid containing mean particle diameter of 36 nm in deionized water as the base fluid. Their results showed the electrical conductivity of alumina nanofluid increased almost linearly with increase in the volumetric concentration of particles. No temperature dependency was studied by them and no correlation for the electrical conductivity of the nanofluid was presented by them. J. Nanosci. Nanotechnol. 11, 1–8, 2011

nf − bf = 3679 049 + 1 085799T − 43 6384 bf

(7)

where T is in C.

3. EXPERIMENTAL SETUP AND PROCEDURE 3.1. Material Preparation In this study three types of nanofluids, namely Al2 O3 , SiO2 and ZnO nanoparticles dispersed in PG/W (60:40) were used. Original concentrated aqueous suspensions of aforementioned nanofluids were procured from Alfa Aesar14 and Nanostructured and Amorphous Materials, Inc.15 The characteristics of materials used in our experiments are tabulated below. From these concentrated parent nanofluids, test samples of different volumetric concentration were prepared. The exact mass of PG/W (60:40) mixture was calculated to attain a desired volumetric concentration of nanofluid. Next, using a precise electronic mass balance, the calculated amount of PG/W (60:40) mixture was added to the concentrated nanofluid by pipettes to reach the exact level of particle volumetric concentration. In this way nanofluid samples starting with 1% volumetric concentration to a maximum of 10% could be prepared. It was only the high concentration of 45 nm Al2 O3 parent fluid with the initial concentration of 50% that yielded a 10% concentration sample. The others yielded sample concentrations of ≤6%. 3.1.1. Sonication The nanofluid samples were subjected to ultrasonication in a Branson Model 551017 sonicator under a frequency of 40 kHz. The ultrasonicator bath was filled with water up to the designated operating level. Then the water was degassed for 5 minutes for removal of dissolved gases as instructed by the manufacturer. Subsequently, the nanofluid sample was placed in the water bath and sonicated for about 3 hours to ensure uniform dispersion of nanoparticles. 3

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where viscosity = AeB/T −To

(5)

Ganguly et al.13 measured the electrical conductivity of Al2 O3 nanoparticles in deionized water of average particle size 13 nm up to a volumetric concentration of 3% within the temperature range of 25–45 C. They expressed that the electrical conductivity of the nanofluid increased with an increase in temperature, and also it increased with an increase in the volumetric concentration. They mentioned that the Maxwell equation developed for electrical conductivity of random suspension of microparticles underpredicted the values when applied to suspensions of nanoparticles. From their experimental data they presented an empirical correlation valid for the Al2 O3 nanofluid in the temperature range of 25 C to 45 C given by


3.3. Benchmark Test Case The benchmark test of the electrical conductivity meter and the probe was performed using the conductivity calibration liquid of Hanna designated as HI 7033. These electrical conductivities measured at various temperatures of the calibration liquid have been plotted in Figure 2. For comparison the conductivity values of this liquid provided by the manufacturer have also been plotted in this figure. It was observed that the measured values of the electrical conductivities and the values published by the manufacturer differed by about ±1%. Thus the benchmark

Electrical conductivity σ μS/cm

The experimental setup, shown in Figure 1, consists of a Hanna HI 452118 conductivity bench meter. The probe HI 76312 is a platinum 4-ring conductivity probe, which is immersed into the liquid to a depth designated by a ring for correct reading. When inserted into the nanofluid, the probe measures the temperature and electrical conductivity of nanofluid in C and S/cm respectively. For measurements above room temperature (around 20 C) the nanofluid was heated to various temperature levels up to 90 C by a heater, VWR Model 320.19 The heater has the capability to stir the liquid with a magnetic stirrer at different rpm to help disperse the nanoparticles uniformly. The stirrer was removed from the beaker when electrical conductivity was being measured to eliminate its effect on readings. The accuracy of the conductivity meter specified by the manufacturer is ±1% of the reading. This conductivity meter and the probe have the capability to measure electrical conductivity from 0 to 10,000 S/cm with settings in four ranges for four different resolutions. The liquid temperature range over which the meter and probe are suitable is −20 C to 120 C. For measurements at temperatures lower than the room temperature, the nanofluid samples were cooled in a temperature controlled freezer (FS 202 Chamber).20 The samples were ultrasonicated for 3 hours before being placed in the freezer chamber. A steady state temperature down to 0 C could be reached within 30 minutes, after which the conductivity measurements were taken.

100 Experimental data Manufacturer’s data

95 90 85 80 75 70 65 60 270

274

278

282

286

290

294

298

302

306

310

Temparature K Fig. 2. Benchmark test case for the electrical conductivity of the calibration solution, Hanna HI 7033.

test showed that the apparatus was working correctly and the measurement procedure was correct. With this confirmation, the electrical conductivities of three nanofluids of varying concentrations, particle size at temperatures ranging from 273K to 363 K (0 C–90 C) were measured. Many of these measurements were repeated to ensure the reproducibility of this data.

4. RESULTS AND DISCUSSION 4.1. Base Fluid Data In the present study, PG/W (60:40) was used as the basefluid to prepare different types of nanofluids. Therefore, the electrical conductivity of this base fluid was measured first, which represented the conductivity values when the nanoparticle concentration is zero. The results of this measurement are shown in Figure 3. A second order polynomial relation fits the data well. bf = 2 316 × 10−5 T 2 − 1 066178 × 10−2 T + 1 27618050

with R2 = 0 9998

(7)

0.5

Electrical conductivity σbf μS\cm

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3.2. Setup

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0.45

PG/W(60:40) Polynomial fit

0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 270

280

290

300

310

320

330

340

350

360

370

Temperature K Fig. 1. Experimental setup for measuring the electrical conductivity of nanofluids.

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Fig. 3. Electrical conductivity of PG/W (60:40) as a function of temperature.

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This equation can be nondimensionalized by the electrical conductivity bfo of the basefluid at a reference temperature To , which was adopted to be the room temperature of 20 C. 2 bf T T = 14 14710791 − 22 22753013 bf0 T0 T0 + 9 080421369

with R2 = 0 9998

(8)

The Eq. (8) is valid in the range 273 K ≤ T ≤ 363 K. 4.2. Aluminum Oxide Nanofluid Figure 4 presents a comparison between the electrical conductivity of the Al2 O3 nanofluid in PG/W (60:40) of three average particle sizes, namely, 10, 20 and 45 nm. Several distinct characteristics are observed. The electrical conductivity of Al2 O3 nanofluids increases as the temperature increases. The electrical conductivity shows a linear variation with temperature. As an example for the 20 nm nanofluid of 1% concentration, the linear variation of electrical conductivity with the temperature is given by Eq. (9). nf = 1 3732T − 355 39

with R2 = 0 9923

(9)

1% Al2O3 10 nm 1% Al2O3 45 nm 2% Al2O3 20 nm 3% Al2O3 20 nm 4% Al2O3 20 nm 0% Al2O3 (PG/W)

1% Al2O3 20 nm 2% Al2O3 10 nm 2% Al2O3 45 nm 3% Al2O3 45 nm 4% Al2O3 45 nm Linear fit

400 300 200 100 0 270

nf = 2 5241T − 641 04

with R2 = 0 99

(10)

valid in the range 273 K ≤ T ≤ 363 K.

Electrical conductivity σnf μS\cm


500

Figure 5 shows the variation in electrical conductivity of the SiO2 nanofluid with temperature. Only one average particle size of 30 nm for this nanofluid was available from the manufacturer. Similar to the observation for Al2 O3 nanofluids, the electrical conductivity of SiO2 nanofluid also increases with an increase in temperature in a linear fashion for a constant concentration, in the temperature range of 273 to 363 K. As an example, the electrical conductivity data of the 3% SiO2 nanofluid matches the expression

450

700 600

4.3. Silicon Dioxide Nanofluid

290

310

330

350

370

Temperature K Fig. 4. Comparison between the electrical conductivity of the Al2 O3 nanofluid as a function of temperature containing 10, 20 and 45 nm particles.


400 350 300

1% SiO2 30 nm 2% SiO2 30 nm 3% SiO2 30 nm 4% SiO2 30 nm 5% SiO2 30 nm 0% SiO2 (PG/W) Linear fit

250 200 150 100 50 0 270

280

290

300

310

320

330

340

350

360

370

Temperature K

Fig. 5. Electrical conductivity variation with temperature for the SiO2 nanofluid in PG/W (60:40) basefluid for particle concentration ranging from 0 to 6%.

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The Eq. (9) is valid in the range 273 K ≤ T ≤ 363 K. The electrical conductivities of 1% Al2 O3 nanofluid of particle sizes of 10, 20 and 45 nm increased by 441%, 444% and 416% respectively, when the temperature increased from 273 K to 363 K (34%). At a specific temperature, as the volumetric concentration of nanofluid increases, its electrical conductivity increases. For example at a temperature of 50 C, the electrical conductivity of the Al2 O3 nanofluid with an average particle size of 45 nm increases by 191% when the volumetric concentration increases from 1% to 4%. At the same temperature of 50 C, the electrical conductivity of this

nanofluid with 20 nm particle size increases by 106%, when the volumetric concentration increases over the same range from 1 to 4%. For about 30% increase in temperature, the nf increases by 400%, but for a 300% increase in , nf increases by 100–200%. Therefore, the variation of electrical conductivity of nanofluid shows a stronger dependence on temperature than the volumetric concentration within the ranges of our experiments. It is observed that for the same particle volumetric concentration, at a fixed temperature, nanofluids containing smaller nanoparticles exhibit higher electrical conductivity than those containing larger nanoparticles. This is due to the presence of more number of charged particles of smaller size for the same volumetric concentration. Comparing the trends of measured data in Figure 4 with the equation Wang and Hirata,7 Eq. (1), we observe an agreement. Their equation predicts that the electrical conductivity of a nanofluid increases with an increase in concentration and a decrease in particle size. Our data exhibit the same trend.


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Again, similar in trend to the Al2 O3 nanofluid, the electrical conductivity of SiO2 nanofluid also increases with an increase in the volumetric concentration of particles. 4.4. Zinc Oxide Nanofluid Figure 6 shows a comparison of the electrical conductivity between two zinc oxide nanofluids containing 36 and 70 nm average particle sizes. Unlike the previous two nanofluids, for this nanofluid the electrical conductivity variation with temperature fits a second order polynomial. As an example, for 1% ZnO nanofluid of 70 nm particle size, the electrical conductivity can be expressed as nf = −0 0012T 2 + 1 0844T − 202 61

with R2 = 0 99 (11)

4.5. Concentration Dependence

160 140 120 100 80

nf = −0 29962 + 12 242 + 3 5475

with R2 = 0 9994 (12)

valid 1% ≤ ≤ 10% In Figure 7, the Maxwell Eq. (4) is plotted for the Al2 O3 nanofluid using the properties data at 20 C at which p value is available from the literature.16 The Maxwell equation clearly underpredicts the electrical conductivity of nanofluids. Similar observation was made by Ganguly et al.13

1% ZnO 70 nm 2% ZnO 70 nm 3% ZnO 70 nm 4% ZnO 70 nm 1% ZnO 36 nm 2% ZnO 36 nm 3% ZnO 36 nm 4% ZnO 36 nm 0% ZnO (PG/W) 2nd order poly. fit

4.6.1. Guidance from Experimental Data From the experimental results analyzed from Figures 4 through 7, it was established that the electrical conductivity of nanofluids were dependent on the particle volumetric concentration, fluid temperature and the nanoparticle diameter. To establish the influence of each of these parameters on the electrical conductivity, the following analyses were conducted by varying each of Table I. Material characteristics of nanofluids used in the present experiments.

Manufacturer

Material

Parent Particle Particle Particle nanofluid electrical size density concentration conductivity16 nm g/cc wt% in H2 O S/cm at 20 C

60 40 20 0 270

280

290

300

310

320

330

340

350

360

370

Temperature K

Fig. 6. Comparison between the electrical conductivity of ZnO nanofluid as a function of temperature of particle sizes 36 and 70 nm.

6

with concentration. For the Al2 O3 nanofluid with 45 nm average particle size, the polynomial fit is

4.6. Development of Correlation

We have presented in Figure 7 the variation of electrical conductivity of Al2 O3 , ZnO and SiO2 nanofluids with volumetric concentration up to 10% for particle sizes ranging from 10 nm to 70 nm. This variation is shown at a constant nanofluid temperature of 50 C, which was around the mean value of the range of temperature over which our measurements were conducted. A second order polynomial fits the data well for the variation of electrical conductivity


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valid in the range 273 K ≤ T ≤ 363 K. From the data it is evident that as the temperature increases, the electrical conductivity of zinc oxide nanofluid also increases. However, it does not increase in the same linear fashion as where the cases for aluminum oxide and silicon dioxide nanoparticles. At higher concentrations and higher temperature range, where the nonlinearity is observed, we repeated these measurements several times with new nanofluid samples to rule out any errors in measurements. In all cases, the measured conductivities where found to be repeatable with differences within ±1 percent.

Fig. 7. Variation of electrical conductivity of three nanofluids with particle volumetric concentration at a mean temperature of 50 C.

Alfa Aesar Al2 O3 Alfa Aesar Al2 O3 Nanostructured

−Al2 O3 and Amorphous Materials, Inc. Nanostructured SiO2 and Amorphous Materials, Inc. Alfa Aesar ZnO Alfa Aesar ZnO

20 45 10

3.614 3.6 3.615

30 50 20

1 02 × 10−10 1 02 × 10−10 1 02 × 10−10

30

2.41515

25

10−9

36 70

5.614 5.6

40 50

— —


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Table II. Regression coefficients of the electrical conductivity correlation (Eq. (16)) for different nanofluids. Regression constants

ZnO

SiO2

Al2 O3

a1 a2 a3 b1 b2 b3 c1 c2 R2 Maximum deviation% Average deviation%

−8177.324 1413.054 2.2848 −2.719 5.594 −2.584 11.681 8.383 0.98 −15% −1.74%

2928.485 23095.615 419.136 −3.373 7.3092 −3.3397 0a 1a 0.98 +10% 3.12%

−1772.883 1128.208 14.425 −2.069 4.578 −2.204 11.456 −16.256 0.98 +20% 3.93%

a

Since only one particle size was tested for SiO2 nanofluid the diameter effect is not included.

those parameters independently. To make the correlation independent of units, the electrical conductivity nf was nondimensionalized by bf using Eq. (8). 4.6.2. Influence of Concentration

conductivity ratio with the particle size, the experimental values of nf /bf of each nanofluid were plotted against its nondimensional average particle sizesd0 /d. The particle size d varied from 10 to 70 nm and d0 , the reference average particle size was adopted to be 100 nm, which is generally accepted as the upper limit for the nanoscale range. From these plots it was determined that variation of the electrical conductivity with an average particle size for each nanofluid followed a linear relation d nf = c1 0 + c2 (15) bf d From these individual analyses of the influence of each parameter, it was concluded that a correlation in the following format would be appropriate for the electrical conductivity. 2 T T nf 2 + b3 = a1 + a2 + a3 b1 + b2 bf T0 T0 d × c1 0 + c2 (16) d Next the statistical package LAB fit21 was used to determine the unknown regression coefficients of Eq. (16) for different nanofluids from their experimental data. The regression coefficients from the statistical analysis are tabulated in Table II. The Eq. (16) with coefficents of Table II has the range of validity; 273 K ≤ T ≤ 363 K; 1% ≤ ≤ 6%; 10 nm ≤ d ≤ 70 nm.

4.6.3. Influence of Temperature

4.7. Comparison of Experimental Values with Correlation

The electrical conductivity nf data of each of nanofluid in Figures 4–6 were nondimensionalized with the basefluid electrical conductivity bf and were plotted against the nondimensionalized temperature T /T0 , for fixed concentration and fixed particle sizes. It was observed that the electrical conductivity ratio nf /bf followed a second order polynomial in T /T0 . A careful analysis into this variation confirmed that although nf was a linear function of T , bf was nonlinear in T as observed in Figure 3. Due to a higher rate of increase of bf with T , the best fit correlation for nf /bf as a function of T /T0 emerged to be of the form 2 T T nf + b3 = b1 + b2 (14) bf T0 T0

Comparison of the electrical conductivity values predicted by the correlation (Eq. (16)) and the experimental values for the ZnO nanofluid is shown as an example in Figure 8 using the software Matlab.22 The central line

4.6.4. Influence of Average Particle Size Figures 4, 6 and 7 had revealed that the electrical conductivity increased as the particle size decreased for fixed temperature and concentration. Therefore, to evaluate this inverse functional nature of the variation of the electrical J. Nanosci. Nanotechnol. 11, 1–8, 2011

Fig. 8. Comparison of the electrical conductivity of ZnO nanofluid values calculated from the present correlation, Eq. (16) with the values obtained from the experiments.

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From Figure 7 and equation Eq. (12) it was observed that the electrical conductivity nf variation as a function of followed a polynomial function of second degree at a constant temperature and particle size. This was tested for each of the nanofluids at several fixed temperatures and particle sizes of 10, 20, 30, 36, 45 and 70 nm. It was found that the nature of the correlation can be represented the best way by nf = a 1 2 + a2 + a3 (13) bf


represents a perfect match between the experimental and correlation values and the two dashed lines are the 95% prediction bound. Only 2 data points out of about 100 data points fall out side the 95% prediction bounds on the upper side. Given a specific concentration, diameter and temperature of this nanofluid within the curve-fit range, the correlation ensures 95% confidence that the electrical conductivity value will be between the lower and upper prediction limits.

5. CONCLUSIONS

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The electrical conductivity of three nanofluids were measured and found to be functions of volumetric concentration, temperature and particle diameter. The electrical conductivity exhibited a strong dependence on temperature. For the Al2 O3 nanofluid an increase of 400% in electrical conductivity was observed for a temperature rise of about 30%. For the same nanofluid, an increase of about 100–200% in electrical conductivity was observed for a volumetric concentration increase of 300%. The electrical conductivity was found to be inversely proportioned to particle diameter. From experimental data a correlation was developed (Eq. (16)) that is suitable for the prediction of the electrical conductivity of these nanofluids. With future research this correlation can be improved to be valid for other nanofluids. Nomenclature d do T To

average particle size, nm reference average particle size, 100 nm absolute temperature, K reference temperature = room temperature, 20 C (293 K)

Greek symbols nf electrical conductivity of nanofluid, S/cm bf electrical conductivity of basefluid, S/cm bfo electrical conductivity of basefluid at reference temperature, S/cm

p

Konakanchi et al.

electrical conductivity of particle material, S/cm volume fraction of nano particles

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Received: 23 September 2010. Accepted: 26 January 2011.

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