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A new effective thermal conductivity model for a bio-nanofluid (blood with nanoparticle Al2O3)
International Communications in Heat and Mass Transfer 37 (2010) 929–934
Contents lists available at ScienceDirect
International Communications in Heat and Mass Transfer j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / i c h m t
A new effective thermal conductivity model for a bio-nanofluid (blood with nanoparticle Al2O3)☆ M. Ghassemi a,⁎, A. Shahidian b, G. Ahmadi c, S. Hamian d a
K.N. Toosi University of Technology, Tehran, Iran Mechanical Engineering Department, K.N. Toosi University of Technology, Tehran, Iran Clarkson University, Potsdam, NY, USA d Mechanical Engineering, K.N. Toosi University of Technology, Tehran, Iran b c
a r t i c l e
i n f o
Available online 18 May 2010 Keywords: Bio-nanofluid Blood Effective thermal conductivity Blood cells
a b s t r a c t Recently application of nano-technology in medicine and cancer therapy has generated a lot of interest in thermal properties of bio-nanofluid such as blood with nanoparticles suspension. In this study effective thermal conductivity of blood with suspension of Al2O3 nanoparticles as a bio-nanofluid was studied. A twostep model based on parallel mixture rule, thermal resistance concept and Maxwell-type equations was developed. First a model based on the parallel mixture rule and thermal resistance concept was used to predict the blood cells thermal conductivity. Then a model for the effective thermal conductivity of the bionanofluid was developed. It was shown that the results of the proposed model for the blood thermal conductivity agree well with the available data in the literature. © 2010 Elsevier Ltd. All rights reserved.
1. Introduction Fluids with suspended nanoparticles are called nanofluids, a term first proposed by Choi in 1995 of the Argonne National Laboratory, U.S.A. [1]. Nanofluid is considered to be the next-generation heat transfer fluids as they offer exciting new possibilities to enhance heat transfer performance compared to pure liquids. Recently, researchers have demonstrated that nanofluids (such as water or ethylene glycol) with CuO or Al2O3 nanoparticles exhibit enhanced thermal conductivity [1]. Thus, the use of nanofluids, for example in heat exchangers, may result in energy and cost savings and should facilitate the trend of device miniaturization. More exotic applications of nanofluids can be envisioned in biomedical engineering and medicine in terms of optimal nano-drug targeting and implantable nano-therapeutic devices [2]. Therefore the knowledge of bionanofluid thermo-physical properties (such as blood with nanoparticles) becomes essential when flow and heat transfer study of blood in drug delivery and new cancer therapy is considered. The enhancement of thermal conductivity achieved in nanofluids is much greater than what has been predicted by conventional theories such as Maxwell [3] or Hamilton and Crosser [4]. Several experimental studies have explained the reason behind the enhancement of effective thermal conductivity such as the effect of the solid/ liquid interfacial layer and the Brownian motion [2,5–9]. For example, ☆ Communicated by W.J. Minkowycz. ⁎ Corresponding author. Mechanical Engineering Department, K.N. Toosi University of Technology, P.O. Box: 19395-1999, Tehran, Iran. E-mail address: [email protected] (M. Ghassemi). 0735-1933/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.icheatmasstransfer.2010.04.010
Xuan and Li [10] summarized all existing experimental observations. They concluded that keff is a function of both thermal conductivities of the nano-material as well as carrier fluid, particle volume fraction, distribution, surface area, and shape. Keblinski et al. [11] listed four possible explanations for the cause of an anomalous increase of thermal conductivity: Brownian motion of the nanoparticles, molecular-level layering of the liquid at the liquid/particle interface, the nature of heat transport in the nanoparticles, and the effects of nanoparticle clustering. They ruled out the possibility of the Brownian motion effect by comparing the time scales of Brownian motion and the thermal response, a point revisited in the Results and discussion section. Xue [5] proposed a thermal conductivity model based on Maxwell's theory and average polarization theory to take care of the interfacial effect (i.e., liquid nano-layer). Bhattacharya et al. [12] investigated the effect of particle Brownian motion by using a molecular dynamics type approach which does not consider the motion of fluid molecules and requires two experimentally determined parameters. Keblinski et al. [13] made an interesting simple review to discuss the properties of nanofluids and future challenges. Most recent studies are about properties of nanofluid with water or ethylene glycol base fluid. The researchers have attempted to find thermo-physical properties of nanofluid especially the effective thermal conductivity in order to analyze thermal behavior of it in different applications. Currently, there is no reliable theory to predict the anomalous thermal conductivity of nanofluids. It is known that the thermal conductivity of nanofluids depends on parameters such as base fluid thermal conductivity, nanoparticles thermal conductivity, the volume
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M. Ghassemi et al. / International Communications in Heat and Mass Transfer 37 (2010) 929–934
plasma as base fluid and Al2O3 as nanoparticles. The second approach does not account for the blood cells. Therefore, Maxwell equation is then used to calculate the thermal conductivity of bio-nanofluid (plasma and nanoparticles and blood cells). In this case, for application of Maxwell's equation, the blood cells are considered as particles, while a mixture of plasma and Al2O3 is the base fluid. (It is known from the literature that Maxwell equation provides good estimate when applied to the fluid with suspended microparticles.)
Nomenclature kf kp klayer keff ϕ (phi) r h
base fluid thermal conductivity nanoparticle thermal conductivity nano-layer thermal conductivity effective thermal conductivity volume fraction nanoparticle radius nano-layer thickness
3. Proposed thermal conductivity model
fraction, the surface area, the shape of the nanoparticles and the temperature [1]. To the authors' knowledge, to date there is no detailed studies on the thermo-physical properties of the blood with suspended nanoparticles. The intention of this study is to develop a new model for effective thermal conductivity of bio-nanofluid (blood + Al2O3). First a model based on the parallel mixture rule [14] and thermal resistance concept [15] was used to predict the blood cells thermal conductivity. Then a model for the effective thermal conductivity of the bio-nanofluid (blood with suspended nanoparticles) was developed. The developed model was based on the Maxwell equation. In addition, the equation developed earlier by Leong et al. [16] was used in the model development. Finally, the new model was used to evaluate the effect of volume fraction of nanoparticles on the blood thermal conductivity.
2. Theory Almost all available theoretical models regarding thermal conductivity are classified into three categories: a) particles are covered with a nano-layer and the thermal conductivity is evaluated for the combined particles and the nano-layer; b) the effective medium theory and the polarization theory are used; and c) the effect of Brownian motion and the other intermolecular forces between particles are considered. The model proposed in this paper is based on the Maxwell equation, parallel mixture rule, and the thermal resistance concept. In addition, the model uses the Leong et al. [16] earlier study which was based on nano-layer coverage assumption. First the model predicts the blood cells thermal conductivity using parallel mixture rule and the thermal resistance concept. Then the model uses the Maxwell and Leong, equations in two different ways to predict the thermal conductivity of bio-nanofluid (blood + Al2O3). Table 1 shows the effective conductivities as suggested by various models. Two approaches were used in the following sections. In the first approach the model equation suggested by Leong et al. [16] is used to calculate the thermal conductivity of bio-nanofluid. Here blood is the base fluid and Al2O3 is the nanoparticle. This is similar to the approach that Leong et al. used to calculate the thermal conductivity of water and Al2O3. In their case water was base fluid and Al2O3 was the particles. In the second approach the results of [16] is used with Table 1 Nanofluid effective thermal conductivity models. Nanofluid thermal conductivity formula keff =
kp + 2kf + 2ðkp −kf Þϕ
k kp + 2kf −ðkp −kf Þϕ f kp −klayer ϕklayer 2β31 −β32 + 1 keff = 3 3 β1 kp + 2klayer − kp −klayer ϕ β1 + β32 −1 3 kp + 2klayer β1 ϕβ32 klayer −kf + kf + 3 β1 kp + 2klayer − kp −klayer ϕ β31 + β32 −1
β = h =r ; β1 = 1 + β =2 ; β2 = 1 + β; klayer = 10kf ðAssumedÞ [16]. r: nanoparticle radius h: nano-layer thickness.
Reference
The proposed model is based on two steps. First we calculate the blood cells thermal conductivity using parallel mixture rule and the thermal resistance concept as described in subsequent Section 3.1. Then the Maxwell and Leong et al. equations are used in two different ways to predict the thermal conductivity of bio-nanofluid (blood + Al2O3), as described in Section 3.2 below. 3.1. Calculation of blood cells thermal conductivity As known blood is a combination of plasma and blood cells including red blood cells (RBC), white blood cells (WBC) and plackets. Table 2 shows the number of concentration and size of blood cells [17]. There are experimental data for thermal conductivity of plasma and red blood cells as well as blood itself. However, the thermal conductivity of the white blood cells and plackets is not known. Therefore, following equations are used to calculate the thermal conductivity of blood cells. 3.1.1. First model According to the parallel mixture rule [14], thermal conductivity of blood is given as: kblood = φplasma kplasma + φblood cells kblood cells :
ð1Þ
All parameters in Eq. (1), with the exception of blood cells thermal conductivity, are known as are listed in Table 3 [18]. Using Eq. (1), the effective thermal conductivity of blood cells is found to be 0.4 W/m K. 3.1.2. Second model This model is based on thermal resistance concept [15]. The model assumes a cubic vessel full of blood with particles completely mixed and considers the heat transfer in the x-direction as illustrated in Fig. 1. As noted before, blood consists of a mixture of plasma and blood cells with the volume fractions of different components are listed in Table 3. The model assumes a one-dimensional steady state heat transfer. The thermal resistance between blood cells (Rblood cells) and plasma (Rplasma) is given as: Rblood = Rcells + Rplasma
ð2Þ
φplasma l 1 φ l = bloodcells + : kblood A kbloodcells A kplasma A
ð3Þ
The values of all parameters in Eq. (3) are listed in Table 3 [18], with the only unknown being the blood cell conductivity. Using the values
Maxwell [4] Table 2 Blood cell concentration and size [17]. Leong et al. [16]
Red blood cells (erythrocyte) White blood cells (leukocyte) Placket (thrombocyte)
Number/mm3
Size (μm)
Cell volume percentage
5 × 106 5 × 103 3 × 105
∼8 ∼ 15 ∼3
97 2 1
M. Ghassemi et al. / International Communications in Heat and Mass Transfer 37 (2010) 929–934 Table 3 Plasma and blood thermal conductivity.
Plasma Blood cells Blood Blood
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Table 4 Fluid and particles data in Maxwell equation.
Volume fraction φ
Thermal conductivity
0.55 0.45 1 1
0.57 W/m K [18] – 0.492 W/m K [18] 0.49–0.55 W/m K [19]
Models
Fluid and particles
1st
Particles (blood cells) 0.45 0.4 (calculated) Fluid (plasma) 0.55 0.57 Whole blood (calculated using Maxwell equation) 0.488 Particles (blood cells) 0.45 0.4215 (calculated) Fluid (plasma) 0.55 0.57 Whole blood (calculated using Maxwell equation) 0.499
2nd
Volume fraction
Thermal conductivity (W/m K)
listed in Table 3, the thermal conductivity of blood cells is found to be 0.42 W/m K. Al2O3 is the nanoparticle. The expression for the effective conductivity of a nanofluid provided by Leong et al. [16] is given as,
3.2. Evaluation of bio-nanofluid effective thermal conductivity As noted before, two approaches are used for evaluating the thermal conductivity of the bio-nanofluid mixture (blood + Al2O3 nanoparticles). 3.2.1. First approach Maxwell equation is used for evaluating the blood thermal conductivity (keff). That is,
keff
kp + 2kf + 2 kp −kf ϕ kf = kp + 2kf − kp −kf ϕ
ð4Þ
In this equation, kp is the particles thermal conductivity (blood cells evaluated from Section 3.1) and kf is the plasma thermal conductivity. Using data that is listed in Table 4, for blood cells thermal conductivity of 0.4, the value of the blood thermal conductivity becomes 0.488 W/m K. This value is about 0.9% lower than the experimental value of 0.492 reported by Holmes [18]. However, for blood cells thermal conductivity of 0.4215 W/m K, the blood thermal conductivity becomes 0.499 W/m K, which is 1.6% higher than the experimental results of Holmes [18]. The predicted values are also listed in Table 4. Finally the result of Leong et al. [16] is used to evaluate the thermal conductivity of bio-nanofluid (keff) where blood is the base fluid and
keff
kp −klayer ϕklayer 2β31 −β32 + 1 = β31 kp + 2klayer − kp −klayer ϕ β31 + β32 −1
ð5Þ
kp + 2klayer β31 ϕβ32 klayer −kf + kf + : β31 kp + 2klayer − kp −klayer ϕ β31 + β32 −1
In Eq. (5) the blood thermal conductivity (kf = kblood) is 0.488 W/m K (according to the first model) or 0.499 W/m K (according to the second model). Also the nanoparticle thermal conductivity is, kp = kAl2O3 = 46 W/m K. The expressions for the other parameters are listed in Table 1, and the nano-layer thermal conductivity is assumed to be given as klayer = 10 ⁎ kf as suggested by Leong et al. [16], where they assumed that nanoparticles are covers by nanolayer. (Here r and h are, respectively, nanoparticle radius and nanolayer thickness.) The block diagram for evaluation of the bionanofluid thermal conductivity is provided in Fig. 2. 3.2.2. Second approach In the second approach, first the mixture of nanoparticles and plasma is considered using the relationships provided by Leong et al. [16], with plasma being the base fluid (kf) and Al2O3 being the nanoparticles (kp). The Maxwell equation as given by Eq. (4) is then used to calculate the thermal conductivity of bio-nanofluid. In Eq. (4) kp is the blood cells thermal conductivity (as described in Section 3.1) and kf is the thermal conductivity of the mixture of plasma and Al2O3, which is calculated by the method suggested by Leong et al. [16]. As noted before, they assumed that the nanoparticles are covers by nanolayer and they assumed that knano-layer = 10 ⁎ kf. The block diagram for this approach is shown in Fig. 3. 4. Results and discussion
Fig. 1. Thermal resistance in a cubic vessel full of blood.
To our knowledge there is no documented experimental data for thermal conductivity of blood with nanoparticles. Therefore, for verification purposes, we compare our calculated blood cells thermal conductivity with red blood cells thermal conductivity reported in the experimental study [19]. Fig. 4 shows that the calculated blood cells thermal conductivity using the first and second models described in Section 3.1, respectively, deviates about 13.3% and 6.7% from the experimental data for the red blood cells (RBC). The predicted effective thermal conductivity of bio-nanofluid (blood with alumina nanoparticle) is compared with the Leong et al. [16] equation in this section. Using the predicted blood thermal conductivity in the first and second models, respectively, 0.488 W/m K and 0.499 W/m K and the experimental blood thermal conductivity of 0.492 W/m K (as the base fluid) and Al2O3 nanoparticles thermal conductivity of 46 W/m K, the results are shown in Fig. 5. In our calculation of the nano-layer thermal conductivity, the
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M. Ghassemi et al. / International Communications in Heat and Mass Transfer 37 (2010) 929–934
Fig. 2. Block diagram for the first approach for evaluating the bio-nanofluid effective thermal conductivity a) using the first model and b) using the second model.
thickness of nano-layer is assumed to be 1 nm and the radius of nanoparticles is 5 nm. The other parameters are described in Table 1. Fig. 5 shows that the prediction of the first and the second models is in good agreement with each other and the earlier model of Leong et al. [16]. The models also properly predicted the expected increasing trend of effective thermal conductivity of nanofluid with increase of volume fraction of nanoparticles. In Fig. 6, the second approach (which was described in Section 3.2) is used. The Leong et.al [16] model presented in Table 1 for nanofluid with water as the base fluid, is used in this figure. Using this model, consider water, plasma and the whole blood as the base fluid and alumina nanoparticles. As noted before, the Maxwell equation is used for calculating the thermal conductivity of bio-nanofluid. (Base fluid is the mixture of plasma and Al2O3 nanoparticles and blood cells are treated as microparticles.) Here the thermal conductivity of the blood cell was evaluated by different methods, and the experimental value provided in [19] was also used. Fig. 6 shows that nanofluid with water/plasma has the highest thermal conductivity. That of blood is lower. But the behavior of blood-based nanofluid resembles the water-based nanofluid. As expected, the effective thermal conductivity of plasma is very close to water. The reason lies on the fact that 91% of plasma is water. Using Maxwell model, the three predicted values are almost identical but
lower than the one predicted by first approach by a factor of 2. These three lines seem to be more compatible with the physics of the problem and can better predict bio-nanofluid effective thermal conductivity. The reason of this behavior is that in the second approach, we use Maxwell theory and consider blood cells microparticles with lower thermal conductivity to the base fluid (suspension contains plasma and nanoparticles). So bio-nanofluid thermal conductivity in this approach is lower than calculated effective thermal conductivity in the first approach. Also we should consider that the size of plasma and nanoparticles is in the same order; the results of equation in Table 1 are adequate, and Maxwell theory has a proper capability in predicting thermal conductivity of suspensions with microparticles. That is why it is appropriate for blood cells. 5. Conclusions Our model is based on two steps. First an intermediate step is utilized. The model uses the parallel mixture rule and the thermal resistance idea and determines the blood cells thermal conductivity. Then we use two approaches to calculate the bio-nanofluid thermal conductivity (blood + Al2O3 nanoparticles). In the first approach, we use Maxwell model to calculate the whole blood thermal conductivity. In the second approach, we use Leong model to determine the thermal
M. Ghassemi et al. / International Communications in Heat and Mass Transfer 37 (2010) 929–934
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Fig. 5. Effective thermal conductivity of bio-nanofluid versus volume fraction of Al2O3 nanoparticles. (Comparison of the proposed first and second models with that of Leong et al. [16]).
Fig. 3. Block diagram for the second approach for evaluating the bio-nanofluid effective thermal conductivity.
conductivity of plasma and nanoparticles. Then by placing the resulted thermal conductivity as kf and gained blood cells thermal conductivity in Section 3.1 as kp in Maxwell equation, bio-nanofluid effective thermal conductivity is determined. Results show that nanofluid with water/plasma has the highest thermal conductivity. That of blood is lower. Using Maxwell model, the three predicted values are almost identical but lower than the one predicted by the first approach by a factor of 2. The results of the second approach seem to be more compatible with the physics of the problem and can better predict bio-nanofluid effective thermal conductivity. The reason of this behavior is that in the second approach, we use Maxwell theory and consider blood cells microparticles with lower thermal conductivity to the base fluid (suspension contains plasma and nanoparticles). So bio-nanofluid thermal conductivity in this approach is lower than calculated effective thermal conductivity in the first approach.
Fig. 4. Blood cells thermal conductivity (calculated and experimental data [19]).
Fig. 6. Effective thermal conductivity of bio-nanofluid versus volume fraction. (Comparisons of the proposed models with that of Leong et al. [16]).
References [1] X. Qi Wang, S. Mujumdar, Heat transfer characteristics of nanofluids: a review, International Journal of Thermal Sciences 46 (2007) 1–19. [2] J. Koo, C. Kleinstreuer, A new thermal conductivity model for nanofluids, Journal of Nanoparticle Research 6 (2004) 577–588. [3] J. Maxwell, A Treatise on Electricity and Magnetism, 2nd edn. Oxford University Press, Cambridge, UK, 1904. [4] R. Hamilton, O. Crosser, Thermal conductivity of heterogeneous two-component systems, I & EC Fundamentals 125 (3) (1962) 187–191. [5] Q.-Z. Xue, Model for effective thermal conductivity of nanofluids, Physics Letters A 307 (2003) 313–317. [6] W. Yu, S.U.S. Choi, The role of interfacial layers in the enhanced thermal conductivity of nanofluids: a renovated Maxwell model, Journal of Nanoparticle Research 5 (1–2) (2003) 167–171. [7] W. Yu, S.U.S. Choi, The role of interfacial layers in the enhanced thermal conductivity of nanofluids: a renovated Hamilton-crosser model, Journal of Nanoparticle Research 6 (4) (2004) 355–361. [8] S.P. Jang, S.U.S. Choi, Appl. Phys. Lett. 85 (2004) 3549. [9] R. Prasher, P. Bhattacharya, P.E. Phelan, Thermal conductivity of nanoscale colloidal solutions (nanofluids), Physical review Letters 94 (2) (2005) 025901. [10] Y. Xuan, Q. Li, Heat transfer enhancement of nanofluids, International Journal oh Heat and Fluid Transfer 21 (2000) 58–64. [11] P. Keblinski, S. Philipot, S. Choi, J. Eastman, Mechanism of heat flow in suspensions of nano-sized particles (nanofluids), Int. J. Heat Mass Transfer 45 (2002) 855–863. [12] P. Bhattacharya, S.K. Saha, A. Yadav, P.E. Phelan, R.S. Prasher, Brownian dynamic simulation to determine the effective thermal conductivity of nanofluids, Journal of Applied Physics 95 (11) (2004) 6492–6494. [13] P. Keblinski, J.A. Eastman, D.G. Cahill, Nanofluids for thermal transport, Materials Today 8 (6) (2005) 36–44. [14] S.K. Das, S.U.S. Choi, Wenhua Yu, T. Pradeep, Nanofluids Science and Technology, John Wiley & Sons, 2008.
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[15] F.P. Incropera, D.P. Dewitt, Introduction to Heat Transfer, Fourth edition, John Wiley & Sons, 2002. [16] K.C. Leong, C. Yang, S.M.S. Murshed, A model of the thermal conductivity of nanofluids — the effect of interfacial layer, Journal of Nanoparticle Research 8 (2006) 245–254. [17] M. Thiriet, Biology and Mechanics of Blood Flows, Part I: Biology, Project-team INRIA-UPMC-CNRS REO, Laboratoire Jacques-Louis Lions, CNRS UMR 7598, Université Pierre et Marie Curie, Springer Science, (2008).
[18] Frank Kreith, D. Yogi Goswami, Bela I. Sandor, Part 4 of The CRC Handbook of Mechanical Engineering: Appendix A “Thermal Conductivity Data for Specific Tissues and Organs for Humans and Other Mammalian Species” by Kenneth. R. Holmes, CRC Press, 2004. [19] V.M. Nahirnyak, S.W. Yoon, C.K. Holland, Acousto-mechanical and thermal properties of clotted blood, J Acoust Soc Am 119 (6) (2006) 3766–3772.
Contents lists available at ScienceDirect
International Communications in Heat and Mass Transfer j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / i c h m t
A new effective thermal conductivity model for a bio-nanofluid (blood with nanoparticle Al2O3)☆ M. Ghassemi a,⁎, A. Shahidian b, G. Ahmadi c, S. Hamian d a
K.N. Toosi University of Technology, Tehran, Iran Mechanical Engineering Department, K.N. Toosi University of Technology, Tehran, Iran Clarkson University, Potsdam, NY, USA d Mechanical Engineering, K.N. Toosi University of Technology, Tehran, Iran b c
a r t i c l e
i n f o
Available online 18 May 2010 Keywords: Bio-nanofluid Blood Effective thermal conductivity Blood cells
a b s t r a c t Recently application of nano-technology in medicine and cancer therapy has generated a lot of interest in thermal properties of bio-nanofluid such as blood with nanoparticles suspension. In this study effective thermal conductivity of blood with suspension of Al2O3 nanoparticles as a bio-nanofluid was studied. A twostep model based on parallel mixture rule, thermal resistance concept and Maxwell-type equations was developed. First a model based on the parallel mixture rule and thermal resistance concept was used to predict the blood cells thermal conductivity. Then a model for the effective thermal conductivity of the bionanofluid was developed. It was shown that the results of the proposed model for the blood thermal conductivity agree well with the available data in the literature. © 2010 Elsevier Ltd. All rights reserved.
1. Introduction Fluids with suspended nanoparticles are called nanofluids, a term first proposed by Choi in 1995 of the Argonne National Laboratory, U.S.A. [1]. Nanofluid is considered to be the next-generation heat transfer fluids as they offer exciting new possibilities to enhance heat transfer performance compared to pure liquids. Recently, researchers have demonstrated that nanofluids (such as water or ethylene glycol) with CuO or Al2O3 nanoparticles exhibit enhanced thermal conductivity [1]. Thus, the use of nanofluids, for example in heat exchangers, may result in energy and cost savings and should facilitate the trend of device miniaturization. More exotic applications of nanofluids can be envisioned in biomedical engineering and medicine in terms of optimal nano-drug targeting and implantable nano-therapeutic devices [2]. Therefore the knowledge of bionanofluid thermo-physical properties (such as blood with nanoparticles) becomes essential when flow and heat transfer study of blood in drug delivery and new cancer therapy is considered. The enhancement of thermal conductivity achieved in nanofluids is much greater than what has been predicted by conventional theories such as Maxwell [3] or Hamilton and Crosser [4]. Several experimental studies have explained the reason behind the enhancement of effective thermal conductivity such as the effect of the solid/ liquid interfacial layer and the Brownian motion [2,5–9]. For example, ☆ Communicated by W.J. Minkowycz. ⁎ Corresponding author. Mechanical Engineering Department, K.N. Toosi University of Technology, P.O. Box: 19395-1999, Tehran, Iran. E-mail address: [email protected] (M. Ghassemi). 0735-1933/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.icheatmasstransfer.2010.04.010
Xuan and Li [10] summarized all existing experimental observations. They concluded that keff is a function of both thermal conductivities of the nano-material as well as carrier fluid, particle volume fraction, distribution, surface area, and shape. Keblinski et al. [11] listed four possible explanations for the cause of an anomalous increase of thermal conductivity: Brownian motion of the nanoparticles, molecular-level layering of the liquid at the liquid/particle interface, the nature of heat transport in the nanoparticles, and the effects of nanoparticle clustering. They ruled out the possibility of the Brownian motion effect by comparing the time scales of Brownian motion and the thermal response, a point revisited in the Results and discussion section. Xue [5] proposed a thermal conductivity model based on Maxwell's theory and average polarization theory to take care of the interfacial effect (i.e., liquid nano-layer). Bhattacharya et al. [12] investigated the effect of particle Brownian motion by using a molecular dynamics type approach which does not consider the motion of fluid molecules and requires two experimentally determined parameters. Keblinski et al. [13] made an interesting simple review to discuss the properties of nanofluids and future challenges. Most recent studies are about properties of nanofluid with water or ethylene glycol base fluid. The researchers have attempted to find thermo-physical properties of nanofluid especially the effective thermal conductivity in order to analyze thermal behavior of it in different applications. Currently, there is no reliable theory to predict the anomalous thermal conductivity of nanofluids. It is known that the thermal conductivity of nanofluids depends on parameters such as base fluid thermal conductivity, nanoparticles thermal conductivity, the volume
930
M. Ghassemi et al. / International Communications in Heat and Mass Transfer 37 (2010) 929–934
plasma as base fluid and Al2O3 as nanoparticles. The second approach does not account for the blood cells. Therefore, Maxwell equation is then used to calculate the thermal conductivity of bio-nanofluid (plasma and nanoparticles and blood cells). In this case, for application of Maxwell's equation, the blood cells are considered as particles, while a mixture of plasma and Al2O3 is the base fluid. (It is known from the literature that Maxwell equation provides good estimate when applied to the fluid with suspended microparticles.)
Nomenclature kf kp klayer keff ϕ (phi) r h
base fluid thermal conductivity nanoparticle thermal conductivity nano-layer thermal conductivity effective thermal conductivity volume fraction nanoparticle radius nano-layer thickness
3. Proposed thermal conductivity model
fraction, the surface area, the shape of the nanoparticles and the temperature [1]. To the authors' knowledge, to date there is no detailed studies on the thermo-physical properties of the blood with suspended nanoparticles. The intention of this study is to develop a new model for effective thermal conductivity of bio-nanofluid (blood + Al2O3). First a model based on the parallel mixture rule [14] and thermal resistance concept [15] was used to predict the blood cells thermal conductivity. Then a model for the effective thermal conductivity of the bio-nanofluid (blood with suspended nanoparticles) was developed. The developed model was based on the Maxwell equation. In addition, the equation developed earlier by Leong et al. [16] was used in the model development. Finally, the new model was used to evaluate the effect of volume fraction of nanoparticles on the blood thermal conductivity.
2. Theory Almost all available theoretical models regarding thermal conductivity are classified into three categories: a) particles are covered with a nano-layer and the thermal conductivity is evaluated for the combined particles and the nano-layer; b) the effective medium theory and the polarization theory are used; and c) the effect of Brownian motion and the other intermolecular forces between particles are considered. The model proposed in this paper is based on the Maxwell equation, parallel mixture rule, and the thermal resistance concept. In addition, the model uses the Leong et al. [16] earlier study which was based on nano-layer coverage assumption. First the model predicts the blood cells thermal conductivity using parallel mixture rule and the thermal resistance concept. Then the model uses the Maxwell and Leong, equations in two different ways to predict the thermal conductivity of bio-nanofluid (blood + Al2O3). Table 1 shows the effective conductivities as suggested by various models. Two approaches were used in the following sections. In the first approach the model equation suggested by Leong et al. [16] is used to calculate the thermal conductivity of bio-nanofluid. Here blood is the base fluid and Al2O3 is the nanoparticle. This is similar to the approach that Leong et al. used to calculate the thermal conductivity of water and Al2O3. In their case water was base fluid and Al2O3 was the particles. In the second approach the results of [16] is used with Table 1 Nanofluid effective thermal conductivity models. Nanofluid thermal conductivity formula keff =
kp + 2kf + 2ðkp −kf Þϕ
k kp + 2kf −ðkp −kf Þϕ f kp −klayer ϕklayer 2β31 −β32 + 1 keff = 3 3 β1 kp + 2klayer − kp −klayer ϕ β1 + β32 −1 3 kp + 2klayer β1 ϕβ32 klayer −kf + kf + 3 β1 kp + 2klayer − kp −klayer ϕ β31 + β32 −1
β = h =r ; β1 = 1 + β =2 ; β2 = 1 + β; klayer = 10kf ðAssumedÞ [16]. r: nanoparticle radius h: nano-layer thickness.
Reference
The proposed model is based on two steps. First we calculate the blood cells thermal conductivity using parallel mixture rule and the thermal resistance concept as described in subsequent Section 3.1. Then the Maxwell and Leong et al. equations are used in two different ways to predict the thermal conductivity of bio-nanofluid (blood + Al2O3), as described in Section 3.2 below. 3.1. Calculation of blood cells thermal conductivity As known blood is a combination of plasma and blood cells including red blood cells (RBC), white blood cells (WBC) and plackets. Table 2 shows the number of concentration and size of blood cells [17]. There are experimental data for thermal conductivity of plasma and red blood cells as well as blood itself. However, the thermal conductivity of the white blood cells and plackets is not known. Therefore, following equations are used to calculate the thermal conductivity of blood cells. 3.1.1. First model According to the parallel mixture rule [14], thermal conductivity of blood is given as: kblood = φplasma kplasma + φblood cells kblood cells :
ð1Þ
All parameters in Eq. (1), with the exception of blood cells thermal conductivity, are known as are listed in Table 3 [18]. Using Eq. (1), the effective thermal conductivity of blood cells is found to be 0.4 W/m K. 3.1.2. Second model This model is based on thermal resistance concept [15]. The model assumes a cubic vessel full of blood with particles completely mixed and considers the heat transfer in the x-direction as illustrated in Fig. 1. As noted before, blood consists of a mixture of plasma and blood cells with the volume fractions of different components are listed in Table 3. The model assumes a one-dimensional steady state heat transfer. The thermal resistance between blood cells (Rblood cells) and plasma (Rplasma) is given as: Rblood = Rcells + Rplasma
ð2Þ
φplasma l 1 φ l = bloodcells + : kblood A kbloodcells A kplasma A
ð3Þ
The values of all parameters in Eq. (3) are listed in Table 3 [18], with the only unknown being the blood cell conductivity. Using the values
Maxwell [4] Table 2 Blood cell concentration and size [17]. Leong et al. [16]
Red blood cells (erythrocyte) White blood cells (leukocyte) Placket (thrombocyte)
Number/mm3
Size (μm)
Cell volume percentage
5 × 106 5 × 103 3 × 105
∼8 ∼ 15 ∼3
97 2 1
M. Ghassemi et al. / International Communications in Heat and Mass Transfer 37 (2010) 929–934 Table 3 Plasma and blood thermal conductivity.
Plasma Blood cells Blood Blood
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Table 4 Fluid and particles data in Maxwell equation.
Volume fraction φ
Thermal conductivity
0.55 0.45 1 1
0.57 W/m K [18] – 0.492 W/m K [18] 0.49–0.55 W/m K [19]
Models
Fluid and particles
1st
Particles (blood cells) 0.45 0.4 (calculated) Fluid (plasma) 0.55 0.57 Whole blood (calculated using Maxwell equation) 0.488 Particles (blood cells) 0.45 0.4215 (calculated) Fluid (plasma) 0.55 0.57 Whole blood (calculated using Maxwell equation) 0.499
2nd
Volume fraction
Thermal conductivity (W/m K)
listed in Table 3, the thermal conductivity of blood cells is found to be 0.42 W/m K. Al2O3 is the nanoparticle. The expression for the effective conductivity of a nanofluid provided by Leong et al. [16] is given as,
3.2. Evaluation of bio-nanofluid effective thermal conductivity As noted before, two approaches are used for evaluating the thermal conductivity of the bio-nanofluid mixture (blood + Al2O3 nanoparticles). 3.2.1. First approach Maxwell equation is used for evaluating the blood thermal conductivity (keff). That is,
keff
kp + 2kf + 2 kp −kf ϕ kf = kp + 2kf − kp −kf ϕ
ð4Þ
In this equation, kp is the particles thermal conductivity (blood cells evaluated from Section 3.1) and kf is the plasma thermal conductivity. Using data that is listed in Table 4, for blood cells thermal conductivity of 0.4, the value of the blood thermal conductivity becomes 0.488 W/m K. This value is about 0.9% lower than the experimental value of 0.492 reported by Holmes [18]. However, for blood cells thermal conductivity of 0.4215 W/m K, the blood thermal conductivity becomes 0.499 W/m K, which is 1.6% higher than the experimental results of Holmes [18]. The predicted values are also listed in Table 4. Finally the result of Leong et al. [16] is used to evaluate the thermal conductivity of bio-nanofluid (keff) where blood is the base fluid and
keff
kp −klayer ϕklayer 2β31 −β32 + 1 = β31 kp + 2klayer − kp −klayer ϕ β31 + β32 −1
ð5Þ
kp + 2klayer β31 ϕβ32 klayer −kf + kf + : β31 kp + 2klayer − kp −klayer ϕ β31 + β32 −1
In Eq. (5) the blood thermal conductivity (kf = kblood) is 0.488 W/m K (according to the first model) or 0.499 W/m K (according to the second model). Also the nanoparticle thermal conductivity is, kp = kAl2O3 = 46 W/m K. The expressions for the other parameters are listed in Table 1, and the nano-layer thermal conductivity is assumed to be given as klayer = 10 ⁎ kf as suggested by Leong et al. [16], where they assumed that nanoparticles are covers by nanolayer. (Here r and h are, respectively, nanoparticle radius and nanolayer thickness.) The block diagram for evaluation of the bionanofluid thermal conductivity is provided in Fig. 2. 3.2.2. Second approach In the second approach, first the mixture of nanoparticles and plasma is considered using the relationships provided by Leong et al. [16], with plasma being the base fluid (kf) and Al2O3 being the nanoparticles (kp). The Maxwell equation as given by Eq. (4) is then used to calculate the thermal conductivity of bio-nanofluid. In Eq. (4) kp is the blood cells thermal conductivity (as described in Section 3.1) and kf is the thermal conductivity of the mixture of plasma and Al2O3, which is calculated by the method suggested by Leong et al. [16]. As noted before, they assumed that the nanoparticles are covers by nanolayer and they assumed that knano-layer = 10 ⁎ kf. The block diagram for this approach is shown in Fig. 3. 4. Results and discussion
Fig. 1. Thermal resistance in a cubic vessel full of blood.
To our knowledge there is no documented experimental data for thermal conductivity of blood with nanoparticles. Therefore, for verification purposes, we compare our calculated blood cells thermal conductivity with red blood cells thermal conductivity reported in the experimental study [19]. Fig. 4 shows that the calculated blood cells thermal conductivity using the first and second models described in Section 3.1, respectively, deviates about 13.3% and 6.7% from the experimental data for the red blood cells (RBC). The predicted effective thermal conductivity of bio-nanofluid (blood with alumina nanoparticle) is compared with the Leong et al. [16] equation in this section. Using the predicted blood thermal conductivity in the first and second models, respectively, 0.488 W/m K and 0.499 W/m K and the experimental blood thermal conductivity of 0.492 W/m K (as the base fluid) and Al2O3 nanoparticles thermal conductivity of 46 W/m K, the results are shown in Fig. 5. In our calculation of the nano-layer thermal conductivity, the
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Fig. 2. Block diagram for the first approach for evaluating the bio-nanofluid effective thermal conductivity a) using the first model and b) using the second model.
thickness of nano-layer is assumed to be 1 nm and the radius of nanoparticles is 5 nm. The other parameters are described in Table 1. Fig. 5 shows that the prediction of the first and the second models is in good agreement with each other and the earlier model of Leong et al. [16]. The models also properly predicted the expected increasing trend of effective thermal conductivity of nanofluid with increase of volume fraction of nanoparticles. In Fig. 6, the second approach (which was described in Section 3.2) is used. The Leong et.al [16] model presented in Table 1 for nanofluid with water as the base fluid, is used in this figure. Using this model, consider water, plasma and the whole blood as the base fluid and alumina nanoparticles. As noted before, the Maxwell equation is used for calculating the thermal conductivity of bio-nanofluid. (Base fluid is the mixture of plasma and Al2O3 nanoparticles and blood cells are treated as microparticles.) Here the thermal conductivity of the blood cell was evaluated by different methods, and the experimental value provided in [19] was also used. Fig. 6 shows that nanofluid with water/plasma has the highest thermal conductivity. That of blood is lower. But the behavior of blood-based nanofluid resembles the water-based nanofluid. As expected, the effective thermal conductivity of plasma is very close to water. The reason lies on the fact that 91% of plasma is water. Using Maxwell model, the three predicted values are almost identical but
lower than the one predicted by first approach by a factor of 2. These three lines seem to be more compatible with the physics of the problem and can better predict bio-nanofluid effective thermal conductivity. The reason of this behavior is that in the second approach, we use Maxwell theory and consider blood cells microparticles with lower thermal conductivity to the base fluid (suspension contains plasma and nanoparticles). So bio-nanofluid thermal conductivity in this approach is lower than calculated effective thermal conductivity in the first approach. Also we should consider that the size of plasma and nanoparticles is in the same order; the results of equation in Table 1 are adequate, and Maxwell theory has a proper capability in predicting thermal conductivity of suspensions with microparticles. That is why it is appropriate for blood cells. 5. Conclusions Our model is based on two steps. First an intermediate step is utilized. The model uses the parallel mixture rule and the thermal resistance idea and determines the blood cells thermal conductivity. Then we use two approaches to calculate the bio-nanofluid thermal conductivity (blood + Al2O3 nanoparticles). In the first approach, we use Maxwell model to calculate the whole blood thermal conductivity. In the second approach, we use Leong model to determine the thermal
M. Ghassemi et al. / International Communications in Heat and Mass Transfer 37 (2010) 929–934
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Fig. 5. Effective thermal conductivity of bio-nanofluid versus volume fraction of Al2O3 nanoparticles. (Comparison of the proposed first and second models with that of Leong et al. [16]).
Fig. 3. Block diagram for the second approach for evaluating the bio-nanofluid effective thermal conductivity.
conductivity of plasma and nanoparticles. Then by placing the resulted thermal conductivity as kf and gained blood cells thermal conductivity in Section 3.1 as kp in Maxwell equation, bio-nanofluid effective thermal conductivity is determined. Results show that nanofluid with water/plasma has the highest thermal conductivity. That of blood is lower. Using Maxwell model, the three predicted values are almost identical but lower than the one predicted by the first approach by a factor of 2. The results of the second approach seem to be more compatible with the physics of the problem and can better predict bio-nanofluid effective thermal conductivity. The reason of this behavior is that in the second approach, we use Maxwell theory and consider blood cells microparticles with lower thermal conductivity to the base fluid (suspension contains plasma and nanoparticles). So bio-nanofluid thermal conductivity in this approach is lower than calculated effective thermal conductivity in the first approach.
Fig. 4. Blood cells thermal conductivity (calculated and experimental data [19]).
Fig. 6. Effective thermal conductivity of bio-nanofluid versus volume fraction. (Comparisons of the proposed models with that of Leong et al. [16]).
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