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Entropy generation analysis of laminar flow of a nanofluid in a circular tube immersed in an isothermal external fluid
Energy 93 (2015) 154e164
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Entropy generation analysis of laminar flow of a nanofluid in a circular tube immersed in an isothermal external fluid Vishal Anand Cyient Limited (Formerly Infotech Enterprises Limited), Plot No.-2, IT Park, Nanakramguda, Manikonda, Gachiboli, Hyderabad, 500032, India
a r t i c l e i n f o
a b s t r a c t
Article history: Received 26 June 2015 Received in revised form 4 September 2015 Accepted 6 September 2015 Available online xxx
This paper is an analytical study of entropy generation in the laminar flow of nanofluids in a circular tube. The tube is immersed in an isothermal external fluid e which is the most general thermal boundary condition but has not been studied in much detail in literature. Two nanofluids, namely e watereAl2O3 and ethylene glycoleAl2O3 have been chosen for this study. The effects of the external Biot number, nondimensional temperature difference and volume fraction on the entropy generation characteristics of the flow have been shown through graphs and the physical reasoning behind the observed trends has been discussed threadbare. It is shown that the addition of nanoparticles is beneficial only at smaller Reynolds number and for less viscous base fluids. Most importantly, it is proved that the entropy generated in the case of a tube immersed in an isothermal external fluid is bounded by those for uniform heat flux and uniform wall temperature boundary conditions. © 2015 Elsevier Ltd. All rights reserved.
Keywords: Isothermal external fluid Entropy generation Convective heat transfer Bejan number Biot number
1. Introduction Enhancement of heat transfer rate in a heat exchanger tube without changing its physical dimensions is an enduring challenge in thermal sciences and engineering. To meet this challenge, several methods are being researched in industry and in academia. These include and are not limited to- making fins on the internal surfaces of tubes, insertion of twisted tapes, introduction of nanoparticles and others. The word “nanofluid” was coined by Choi [1]. Nanofluid is a suspension of nanoparticles in a base fluid. The principal way in which nanofluids increase the performance of a thermal system is by enhancing the thermal conductivity of the fluid. To that end, a body of work is devoted to investigation of the thermophysical properties of nanofluids as functions of their size and concentration [2e5]. A study by Xuan and Li [6] measured the thermal conductivity of copper based nanofluid. Lee et al. [7] measured the thermal conductivities of oxide nanofluids experimentally. Das et al. [8] included the effect of temperature on the thermal conductivity of water based nanofluids. There have been other models of thermophysical properties presented by researchers like Masoumi et al. [9], Corcione [10] and Khanafer and Vafai [11]. Apart from the modeling of thermophysical properties, there has been considerable research to understand the effect of
E-mail address: [email protected]. http://dx.doi.org/10.1016/j.energy.2015.09.019 0360-5442/© 2015 Elsevier Ltd. All rights reserved.
nanofluids on the heat transfer rate itself. This research has been performed in both experimental and numerical domains. Pak and Cho [12] proposed a correlation for Nusselt number for turbulent convection of nanofluids. Similar correlation was proposed for copper based nanofluids by Xuan and Li [13]. As far as numerical investigation is considered, Maiga et al. [14] studied the laminar and turbulent heat transfer of nanofluids using the commercially available code FLUENT. Bahiraei and Hangi [15] numerically investigated the hydrothermal characteristics of nanofluid in a C shaped chaotic channel using both single-phase and two-phase methods. Bahiraei [16] used the two-phase EulereLagrange method to find the phenomenological constants needed to calculate the diffusion fluxes in nanofluid laminar flow in a circular pipe. A very recent line of research in this domain pertains to magnetic nanofluids e which comprise of non-magnetic base fluid and magnetic nanoparticles. Here, mass transfer and heat transfer can be controlled by applying magnetic fields. Some of the relevant research work in this area can be accessed in Refs. [17,18]. A prominent disadvantage concerning the use of nanofluids is that due to the concomitant increase in viscosity, the pressure drop (and thereby, the pumping power required) is higher. This means that the heat transfer engineer/researcher should continuously -vis the evaluate the advantage of high heat transfer rate vis-a penalty of high pumping power requirement. An effective tool for this evaluation is the entropy generation analysis. Entropy generated due to heat transfer decreases while the entropy generated due to pressure difference increases with introduction of
V. Anand / Energy 93 (2015) 154e164
Nomenclature A Be Bi Cp D Ec f FF h b h he k L m_ Ns,a Nu b Nu P Pr q_ 00 _ q0 Q_ r Re s S_ St
cross-sectional area (m2) Bejan number Biot number he r=k specific heat capacity (J/kg K) cross sectional diameter (m) Eckert number (defined in Eq. (30)) friction factor entropy generated due to fluid friction internal heat transfer coefficientq_ 00 =ðTw TÞ (W/m2K) overall heat transfer coefficient q_ 00 =ðTe TÞ (W/m2K) external heat transfer coefficient q_ 00 =ðTe Tw Þ (W/m2K) thermal conductivity (W/m K) duct length (m) mass flow rate (kg/s) entropy generation augmentation number internal Nusselt number hD/k overall Nusselt number b hD/k pressure (Pa) Prandtl number local heat flux (W/m2) heat rate per unit length (W/m) heat rate (W) radius of tube (m) Reynolds number specific entropy (J/kg K)
T U
155
temperature (K) average fluid velocity (m/s)
Greek letters a dimensionless parameter (defined in Eq. (28)) t dimensionless temperature difference (defined in Eq. (29)) l dimensionless length (defined in Eq. (15)) lL dimensionless length (defined in Eq. (17)) j dimensionless entropy generation rate (defined in Eq. (31)) 4 volume fraction of nanoparticles DP pressure drop (Pa) r fluid density (kg/m3) m fluid viscosity (Ns/m2) p1 dimensionless parameter (defined in Eq. (33)) p2 dimensionless parameter (defined in Eq. (34)) Subscripts avg average bf base fluid e external i internal nf nanofluid np nanoparticle out outlet ref reference w wall
entropy rate (W/K) Stanton number (defined in Eq. (13))
nanoparticles in the fluid. Thus, it is possible e indeed desirable, to gauge the effectiveness of nanofluids by studying their total entropy generation rates. The entropy generation of nanofluids between two co-rotating cylinders was investigated by Mahian et al. [19]. They showed that it is possible to minimize the entropy generated in the nanofluid with respect to the volume fraction of the nanoparticles. Bianco et al. [20] studied the entropy generated in nanofluid under turbulent conditions in a circular tube subject to constant wall heat flux. They showed that the optimum nanoparticle concentration for minimum entropy generation is lower at higher Reynolds number. Leong et al. [21] studied the effect of cross-sectional shape on the entropy generation analyses of nanofluids. They found out that the circular tube generated the least amount of entropy compared to the tubes of other crosssections. The same research group also studied the entropy generation in a tube with uniform wall temperature for nanofluids derived from alumina and titanium dioxide [22]. They concluded that titanium oxide nanofluids generate less entropy than that generated by alumina based nanofluids. It is to be noted that all the cases of entropy generation analysis considered in extant literature have been based on one of the two common thermal boundary conditions: UWT (uniform wall temperature) and UHF (uniform heat flux). Sparrow and Patankar in a seminal study in 1977 [23] showed that these two boundary conditions are special cases of a more generic boundary condition, namely e immersion in an isothermal external fluid. When the thermal contact between the isothermal external fluid and the tube wall is superior, UWT boundary condition is obtained. On the other hand, if the external thermal contact is poor, the boundary condition reduces to UHF. The entropy generation analysis for laminar flow in a circular tube immersed in an isothermal external fluid was performed by
the author in an earlier publication [24], but that analysis was only for a single phase fluid. The concept of heat transfer enhancement by the introduction of nanoparticles was outside the scope of that paper e but is included in the current paper. This paper deals with entropy generation analyses of nanofluid flow in a circular tube immersed in an isothermal external fluid. Two nanofluids: watereAl2O3 and ethylene glycoleAl2O3 have been chosen for the study. These two nanofluids have been chosen primarily because they are the most commonly used nanofluids in industry and in research. Moreover, the fact that the difference in these two nanofluids is only due to the difference in base fluid e the nanoparticles are the same e makes the assessment more systematic and organized. The novelty of this paper derives from the choice of the thermal boundary condition, namely e immersion in an isothermal external fluid. This is the most common thermal boundary condition seen in nature as well as in industry. For example, the thermal characteristics of a pipe carrying water in ambient surroundings, an oil pipeline in the ocean-bed, and heat exchanger tubes in shell-and-tube heat exchangers will be governed by this thermal boundary condition. It is shown in the results section that both UHF and UWT boundary conditions serve as lower and upper bounds respectively for the isothermal external fluid boundary condition and are thus only special cases of a more generic boundary condition. 2. Problem statement The schematic of the physical system under consideration is shown in Fig. 1. It consists of a circular tube immersed in an isothermal external fluid, which is at uniform and constant temperature Te. The diameter of the tube is D. The nanofluid (watereAl2O3 or ethylene glycoleAl2O3) enters the tube at temperature Ti.
156
V. Anand / Energy 93 (2015) 154e164
These equations were proposed by Maiga et al. [14] and are derived from the experimental works carried out by Lee et al. [7] and Choi et al. [28]. Other researchers in nanofluid research have also used the same correlations for watereAl2O3 and ethylene glycoleAl2O3 nanofluids [29].
3.2. Entropy generation analysis
Fig. 1. The schematic of the physical system under study. Also shown is an infinitesimal control volume.
The flow is steady, laminar and hydrodynamically and thermally fully developed. The thermophysical properties of the base fluids and the nanoparticles have been taken as constants.
3.1. Thermophysical properties of nanofluids
(1)
Here np refers to nanoparticle, bf refers to base fluid, nf refers to nanofluid, while 4 refers to the volume fraction of nanoparticles in the fluid. The specific heat of the nanofluid is evaluated by using the following formula:
Cp;nf
2
mnf ¼ 1234 þ 7:34 þ 1 mbf knf
¼ 4:9742 þ 2:724 þ 1 kbf
(2)
(3) (4)
For ethylene glycoleAl2O3
2
(8)
where b h is the overall heat transfer coefficient, between the external fluid and the internal fluid. Solving the two equations above for T(x), we obtain:
TðxÞ ¼ Te ðTe Ti Þe
b
4 h x rUDCp
(9)
Here, U is the average velocity of the flow. The above equation can be written in terms of non-dimensional parameters as:
TðxÞ ¼ Te ðTe Ti Þe
_
Nu 4Re:Pr l
¼ Te ðTe Ti Þeð4StlÞ
(10)
where, Re is the Reynolds number based on tube diameter:
This formula is based on the assumption that the nanoparticle and base fluid are at thermal equilibrium; there is no heat exchange between a nanoparticle and its surrounding base fluid and they are at same temperature. This formula was presented first by Buongiorno [26] and has been validated experimentally by Pak and Cho [12] and Xuan and Roetzel [27]. The viscosity and thermal conductivity of the nanofluids are calculated using the following formulae: For watereAl2O3:
(7)
b q_ 0 ¼ hpDðT e TÞ
Any theoretical analysis of nanofluids behavior leans heavily on the models used to determine their thermophysical properties as a function of constituents' properties and concentration. For an understanding of how modeling of properties of nanofluids affects their entropy generation analyses, the reader is directed to Singh et al. [25]. The density of the nanofluid is evaluated by using the following formula which is commonly employed for mixtures:
ð1 4Þrbf Cp;bf þ 4rnp Cp;np ¼ rnf
_ p dT ¼ q_ 0 dx mC
Here, Cp is the specific heat of the fluid and m_ is the mass flow rate. q_ 0 is the rate of heat transfer per unit length from the external fluid to the internal fluid and is given by
3. Mathematical formulation
rnf ¼ ð1 4Þrbf þ 4rnp
The entropy generation analysis of laminar flow in a tube immersed in an isothermal, external fluid has been carried out by this author in an earlier publication [24] and is presented here in brief: Consider a CV (control volume) of infinitesimal length dx, in the flow. The energy balance for this CV, (neglecting axial conduction), is given by:
mnf ¼ 3064 0:194 þ 1 mbf
(5)
knf ¼ 28:90542 þ 2:82734 þ 1 kbf
(6)
Re ¼
rUD m
(11)
b is the overall Nusselt number, given as: Here, Nu
b b ¼ hD Nu k
(12)
St is the Stanton number, given by:
St ¼
b b h Nu ¼ rUCp Re:Pr
(13)
r is the fluid density.Pr is the Prandtl number:
Pr ¼
mCp k
(14)
m is the fluid viscosity and k is the thermal conductivity of the fluid. And l is the non-dimensional duct length, given by:
l¼
x D
The temperature at the outlet of the duct Tout is given by
(15)
V. Anand / Energy 93 (2015) 154e164
Tout ¼ TðLÞ ¼ Te ðTe Ti Þeð4StlL Þ
(16)
where,
L lL ¼ D
Tout ¼ TðLÞ ¼ Te ðTe Ti Þe
b
a¼1
N ukpL _ p mC
(18)
The rate of total heat transferred to this CV is given by
_ p ðTout Ti Þ Q_ ¼ mC _ p Te Ti ðTe Ti Þe4StlL ¼ mC
(19)
dS_gen ¼
q_ 0 dx _ þ mds Tw
Here Tw is the temperature of the wall and ds is the infinitesimal entropy change across this CV. Now,
dS_gen ¼
b e TÞpDdx hðT _ þ mds b hðTe TÞ þ T h
(22)
For an incompressible fluid:
_ ¼ m_ mds
Cp dT dP m_ rT T
(23)
dP ¼ dx 2D
_ p dT mfU _ 2 mC dx þ 2DT T
(25)
Substituting in the Eq. (22) we get:
b hðTe TÞpDdx dT fU 2 _ p þ m_ þ mC dx ¼ b T 2DT hðTeTÞ þ T h
j¼
1 te4StlL fEct e4StlL t 1 1 tae4StlL þ ln ln þ ln a 8St 1t 1 ta 1t (31)
The significance of dimensionless entropy generation is that it has been non-dimensionalized by using mass flow rate (and specific heat). This allows the researcher to make a valid comparison of the non-dimensional entropy generation of flows with different mass flow rates. The Eq. (31) can also be written as function of Re in place of St ! b Nu St ¼ RePr
1 tep1 =Re 1 1 taep1 =Re ln þ ln a 1 ta 1t p =Re e 1 t þ p2 Re2 ln 1t
Substitute the value of T(x) and integrate from x ¼ 0 to x ¼ L, we obtain the total entropy generated in the duct as:
(32)
where
p1 ¼
b 4 Nul Lk mCp
(33)
p2 ¼
ðf ReÞm3 b e ðDrÞ2 k 8 NuT
(34)
Bejan number: Bejan number is defined as:
Be ¼ (26)
(30)
S_gen _ p mC
(24)
Thus we get:
(29)
In literature, dimensionless entropy generation has been used to quantify the entropy generated. It is given by:
j¼
where the pressure drop per unit length is related to friction factor ( f ) as:
f rU 2
U2 Cp ðTe Ti Þ
(21)
Substitute the values of q_ 0 and Tw from Eq. (21) into Eq. (20), the following expression for infinitesimal entropy generation rate is obtained:
(28)
Ec is the Eckert number:
¼
b q_ 0 ¼ hpDðT e TÞ ¼ hpDðTw TÞ b hðTe TÞ þT 0Tw ¼ h
Te Ti Te
Ec ¼ (20)
b Nu Nu
where, Nu is the internal Nusselt number ¼ hD/k.t is the nondimensional temperature difference, defined as:
t¼
The entropy generated in this infinitesimal CV is given by:
dS_gen
(27)
(17)
Writing Eq. (16) in terms of mass flow rate, we obtain:
_ ¼ mds
" 1 te4StlL 1 1 tae4StlL _ p ln þ ln S_gen ¼ mC 1t a 1 ta # fEct e4StlL t þ ln 8St 1 t
157
EntropyGeneratedDueToHeatTransfer TotalEntropyGenerated
At Be ¼ 1, all the entropy generated is due to heat transfer. At Be ¼ 0, all the entropy generated is due to fluid friction. In the current analysis, the entropy generated due to fluid friction is given by:
158
V. Anand / Energy 93 (2015) 154e164
FF ¼ lim j ¼ p1 p2 Re Te /Ti
(35)
where FF ¼ Entropy generated due to fluid friction only. So,
Be ¼
j FF FF ¼1 j j
(36)
Entropy augmentation number: To measure the efficacy of any heat transfer augmentation technique, its performance must be compared with respect to the original system. Since the current analysis deals with entropy generation, the relevant augmentation number is the entropy augmentation number, defined as:
Ns;a ¼
Entropy Generated in System with nanofluids Entropy Generated in System with basefluids only
Table 2 b as a function of Bi. Nu and Nu Bi
Nu
b Nu
0 0.1 0.25 0.5 1 2 5 10 100 ∞
4.364 4.330 4.284 4.221 4.122 3.997 3.840 3.758 3.663 3.657
0 0.1909 0.4471 0.8075 1.345 1.998 2.773 3.163 3.597 3.657
Taken from Ref. [23].
Table 3 Different parameters used in the analysis.
(37) By this definition, a values of Ns,a lower than 1 is desirable. Only then, the introduction of nanoparticles is a thermodynamically viable option. In this analysis based on the above equations, the properties of the base fluids are taken from Incropera et al. [30] while those of Al2O3 have been taken from Ref. [21]. They have been presented in Table 1. For different values of volume concentration of Al2O3, these properties are used to calculate the thermophysical properties of the nanofluid using the equations presented in Section 3. b as functions of external Biot number Bi The values of Nu and Nu have been taken from Ref. [23] and presented in Table 2. The value of friction factor for laminar flow in a tube is f ¼ 64/Re from Ref. [30]. Similarly, the numerical values of the other parameters used in the analysis have been tabulated in Table 3. These values are commensurate with those used elsewhere in archival literature. 4. Results The problem statement has been presented in Section 2. Pursuant to the theme of this study, the constitutive equations for the thermophysical properties of nanofluids have been presented in Section 3. The expressions for the non-dimensional entropy generation rate, Bejan number and non-dimensional entropy augmentation number have been derived in the later part of Section 3. Now, in this section, we proceed to present the results and analyze the trends observed therein. In this section, the results have been presented for both water and ethylene glycol. A comparison has also been made between the results of isothermal external fluid boundary condition presented here and those for the more common thermal boundary conditions of uniform wall temperature and uniform heat flux. 4.1. WatereAl2O3 a. Effect on j: For the nanofluid combination of watereAl2O3, the variation of non-dimensional entropy generation rate j with Re is shown in Fig. 2. The effect of Biot number is captured in Fig. 2(a), while Fig. 2(b) and (c) show the effects of the volume
Parameters
Numerical values
Te D L
310 K 0.08 m 1.0 m
concentration 4 and non-dimensional temperature difference t respectively. From Fig. 2, it is observed that the non-dimensional entropy generation rate j decreases with increase in Re. The reason for this trend is attributed to the fact that as the Reynolds number increases, the temperature at any point in the tube tends to the inlet temperature ðfrom Eq: 10 lim TðxÞ ¼ Ti Þ, an increase in Reynolds Re/∞ number causes the temperature gradients inside the fluid to decrease; the whole fluid control volume tends to a uniform temperature. As a result of the decrease in temperature gradients, the entropy generated inside the fluid due to heat transfer decreases. On the other hand, an increase in Reynolds number causes the entropy generated due to fluid friction to increase. But, as shown later in Section 4.1.b, the Bejan number for watereAl2O3 nanofluid is close to 1. This indicates that the entropy generated due to heat transfer is much greater than the entropy generated due to fluid friction and thus the overall entropy generation rate is influenced primarily by the trends in entropy generation due to heat transfer. Fig. 2(a) shows that the non-dimensional entropy generation rate j increases with increase in Biot number Bi. This trend is explained by appreciating that an increase in Bi means that the thermal resistance between the isothermal external fluid and the wall of the tube decreases. The heat transfer rate to the nanofluid is thus enhanced and temperature gradients inside the fluid increases. This translates into higher entropy generation rate. Fig. 2(b) shows that the entropy generation rate decreased with rise in volume fraction 4. To understand this trend, we recognize that the motive of adding nanoparticles is to augment the heat transfer rate by improving the thermal conductivity of the nanofluid. The increase in thermal conductivity leads to decrease in temperature gradients inside the fluid which reduces the entropy generated due to heat transfer. On the other hand, increase in volume fraction 4 also leads to increase in viscosity of the nanofluid
Table 1 Properties of the base fluid and nanoparticles.
Water Ethylene glycol Al2O3
Thermal conductivity (W/m K)
Thermal specific Heat (J/kg K)
Viscosity (Ns/m2)
Density (kg/m3)
0.613 0.252 40
4180 2415 880
0.000855 0.0157
1000 1114.4 3900
V. Anand / Energy 93 (2015) 154e164
a.)
a.)
10
1 0.9999 0.9998 0.9997 0.9996 0.9995 0.9994 0.9993 0.9992 0.9991
Bi=1.0
8
Bi=2.0
6
Bi=5.0
4
Bi=10.0
Bejan number, Be
Non-dimensional entropy 6 generation, ψ x10
12
159
2 0
Bi=1.0 Bi=2.0 Bi=5.0 Bi=10.0
0
0
2000
4000
1000
2000
3000
4000
5000
Reynolds number, Re
6000
Reynolds number, Re 1
b.)
b.)
0.9999 volume_frac on=0.01
3.5
volume_frac on=0.02
volume_frac on=0.03
3
volume_frac on=0.04
2.5
volume_frac on=0.05
volume_frac on=0.01
0.9996
volume_frac on=0.02
0.9995
volume_frac on=0.03
0.9994
volume_frac on=0.04
1.5
0.9993
volume_frac on=0.05
1
0.9992 500
0.5
1500
2500
3500
4500
Reynolds number, Re
0 1500
2500 3500 Reynolds number, Re
4500
5500
16 Non-dimensional entropy generation, ψ x10
0.9997
2
500 c.)
0.9998 Bejan number, Be
4
14
Non-dimensional temperature difference=0.005
12
Non-dimensional temperature difference =0.01
10 8
Non-dimensional temperature difference =0.02
6
c.)
1.0002 1
Bejan number, Be
Dimensionless Entropy Generation, ψ x10
4.5
0.9998
Non-dimensional temperature difference=0.005 Non-dimensional temperature difference =0.01 Non-dimensional temperature difference =0.02
0.9996 0.9994 0.9992 0.999 0.9988 0.9986 0
1000
2000
3000
4000
5000
Reynolds number, Re
4
Fig. 3. Variation of Bejan number with Reynolds number for watereAl2O3 for different values of a.) Bi b.) volume fraction 4 and c.) non-dimensional temperature difference t.
2 0 0
2000
4000
6000
Reynolds number, Re Fig. 2. Variation of dimensionless entropy generation j with Reynolds number for watereAl2O3 for different values of a.) Bi b.) volume fraction 4 and c.) non-dimensional temperature difference t.
and a corresponding increase in entropy generated due to fluid friction. But as mentioned earlier, for this nanofluid e watereAl2O3, the entropy generated due to fluid friction is miniscule, so the total entropy generation rate follows the trend of entropy generated due to heat transfer. And thus, entropy generation rate decreases with addition of nanoparticle. Fig. 2(c) shows that the entropy generation increases with increase in non-dimensional temperature difference on account of increase in heat transfer rate and the consequent increase in temperature gradients inside the fluid. Effect on Be: Fig. 3 shows the variation of Be with Re for watereAl2O3 nanofluid. The effects of Bi, 4 and t on Bejan number have been captured in Fig. 3(a)e(c) respectively. The first observation is that for this nanofluid, Bejan
number is close to 1 for all the values of parameters considered. This leads to the conclusion that for a mildly viscous base fluid like water, the entropy generated due to heat transfer is overwhelmingly high, irrespective of the volume fraction of nanoparticle. We also observe that Bejan number decreases with increase in Re. This is attributed to the twin reasons of increase in entropy generated due to fluid friction at high Re, and decrease in entropy generated due to heat transfer because of decline in temperature gradients inside the fluid. It is seen from Fig. 3(a) that Be increases with increase in Bi. This trend is attributed to the increased heat transfer rate to the nanofluid at higher Bi, which increases the temperature gradients inside the fluid and consequently the entropy generated due to heat transfer increases. Fig. 3(b) shows that an increase in volume fraction 4 leads to a decrease in value of Bejan number. To explain this trend, we recognize that an increase in volume fraction increases the thermal conductivity of the nanofluid. This brings down the temperature gradients inside the fluid and thereby reduces the entropy generated due to heat transfer. Further to this, the entropy generated due
V. Anand / Energy 93 (2015) 154e164
to fluid friction also increases because of increase in viscosity. These two phenomena work in tandem to bring down the value of Bejan number. Fig. 3(c) shows that Be increases with increase in t on account of enhanced heat transfer.
a.)
b. Effect on Ns,a: As mentioned earlier, an important parameter to measure the performance of any heat transfer augmentation technique is the augmentation number which quantifies the -vis that of thermal performance of the augmented system vis-a the original system. Since this study deals with second law analysis of nanofluids, the relevant augmentation number is the entropy augmentation number, defined as the ratio of entropy generated in nanofluids over the entropy generated in base fluids (see Eq. (37)). By this definition, a value of Ns,a lower than 1 is more desirable.
4.2. Ethylene glycoleAl2O3 Next we consider another nanofluid e a combination of ethylene glycol and Al2O3. As seen from Table 1, the viscosity of ethylene glycol is about 400 times that of water. In this section, we will understand how this disparity in viscosity (between water and ethylene glycol) affects the entropy generation characteristics of ethylene glycoleAl2O3 nanofluid. a. Effect on j: The variation of j with Reynolds number for ethylene glycoleAl2O3 nanofluid is shown in Fig. 5. The effects of Bi, 4 and t on j are rendered through Fig. 5(a)e(c) respectively. From Fig. 5(a), it is seen that the value of j first decreases and then increases with increase in Re- especially at low values of Bi (Bi ¼ 1,2). This is explained as follows. Total entropy generation rate is the sum of entropy generation rate due to heat transfer and that due to fluid friction. Initially, j decreases with increase in Re, because an increase in Re decreases the temperature gradients inside the fluid, thereby decreasing the heat transfer irreversibility. But at lager Re, due to the high viscosity of ethylene glycol, the entropy generated due to fluid friction increases and compensates for the decline in entropy generated due to heat transfer. This
0.946
Bi=1.0
0.945
Bi=2.0
0.944
Bi=5.0
0.943
Bi=10.0
0.942 0.941 0.94 0.939 0
2000
4000
6000
Reynolds number, Re
Ns,a
b.)
0.99 0.97 0.95 0.93 0.91 0.89 0.87 0.85 0.83 0.81 0.79
volume_frac on=0.01 volume_frac on=0.02 volume_frac on=0.03 volume_frac on=0.05 voulme_frac on=0.04
0
2000
4000
6000
Reynolds number, Re
c.)
0.945
Non-dimensional temperature difference=0.005
0.944
Non-dimensional temperature difference =0.01
0.943 Ns,a
Fig. 4 depicts the variation of Ns,a with Reynolds number for the watereAl2O3 nanofluid. The effects of Bi, 4 and t on Ns,a have been shown in Fig. 4(a)e(c) respectively. The foremost observation from these figures is that Ns,a is less than 1 for all values of Re, Bi, t and 4. Thus, from the standpoint of the second law, addition of nanoparticles like Al2O3 to water always improves the thermodynamic performance of the heat exchange tube. We also notice that the value of Ns,a keeps decreasing with increase in Re. This is attributed to the fact that an increase in Re causes the entropy generated due to heat transfer to decrease and this decrease in entropy generation is accentuated by the introduction of the nanoparticles. From Fig. 4(a), we observe that Ns,a is the least for smallest Bi, all other parameters kept same. This is explained as follows. For Bi ¼ 1, the heat transfer rate to the fluid is the least and the temperature field inside the fluid is the weakest. The introduction of nanoparticles is thus able to make a stronger impact on the temperature field inside the fluid, reducing the temperature gradients by improving the thermal conductivity. This in turn reduces the entropy generation rate by a higher amount. Fig. 4(b) shows that Ns,a decreases with increase in volume fraction 4. This trend is also along expected lines as introduction of higher concentration of Al2O3 leads to steeper decline in entropy generated due to heat transfer by improving thermal conduction inside the fluid.
0.947
Ns,a
160
0.942
Non-dimensional temperature difference =0.02
0.941 0.94 0.939 0
2000
4000
6000
Reynolds number, Re Fig. 4. Variation of entropy augmentation number Ns,a with Reynolds number for watereAl2O3 for different values of a.) Bi b.) volume fraction 4 and c.) non-dimensional temperature difference t.
results in net increase in total entropy generation rate. This trend is more apparent at lower values of Bi, because at lower values of Bi, the entropy generation due to heat transfer is even lower. Fig. 5(b) shows the variation of j with Re for different values of 4. It can be discerned from the figure that at low values of Reynolds number, j is higher for smaller 4, while at higher values of Reynolds number, j is higher for larger 4. This can be explained as follows. At low Re, entropy generated due to heat transfer is higher as opposed to the situation at high Re e where entropy generated due to fluid friction dominates. Entropy generated due to heat transfer decreases with increase in 4, while entropy generated due to fluid friction increases with 4. This accounts for decrease in total entropy generation rate with 4 at lower Re and the corresponding increase at higher Re.
V. Anand / Energy 93 (2015) 154e164
velocity of water will be less than that of ethylene glycol (due to disparity in value of viscosity). So the temperature gradients inside water are higher and thus entropy generated due to heat transfer is also high. This leads to total (dimensionless) entropy generation being higher for water than that for ethylene glycol.
Non dimensional entropy genera on, ψ x 106
100
10
Bi=1
Bi=2
Bi=5
Bi=10
b. Effect on Be: The variation of Be with Reynolds number is rendered in Fig. 6. Fig. 6(a) captures the effect of Bi, while the effects of 4 and those of t are presented in Fig. 6(b) and (c) respectively.
1 0
1000 2000 Reynolds number, Re
3000
0.1
It is seen from the figure that Be decreases with increase in Reynolds number. This is along expected lines as an increase in Re leads to decrease in heat transfer irreversibility while at the same time it accentuates the fluid friction irreversibility. The most important observation is that Bejan number falls below 0.5 for high value of Re for certain values of Bi, 4 and t. This is quite contrary to the trend observed in water, where Be remained close to 1. This happens because ethylene glycol is much more viscous than water, so the entropy generated due to fluid friction is very high at higher Re. The lowering of Bejan number below 0.5 is more likely to occur
0.01 a.)
10
volume_frac on=0.02
Non-dimensional entropy genera on, ψ x 106
volume_frac on=0.03 volume_frac on=0.04 volume_frac on=0.05
1
500
1000
1500
2000
2500
3000
3500
a.)
1.2
1 Bejan number, Be
0
0.1
b.)
0.01
Bi=1
Bi=2
Bi=5
Bi=10
0.8 0.6 0.4 0.2 0 0
non-dimensional temperature difference=0.005
500
1000
1500
2000
2500
3000
Reynolds number, Re
non-dimensional temperature difference =0.01
10
1
b.) non-dimensional temperature difference =0.02
volume_frac on=0.02
0.9
volume_frac on=0.03
0.8
1
10
510
1010
1510
2010
2510
Bejan number, Be
Non-dimensional entropy genera on, ψ x106
100
0.1
0.7
volume_frac on=0.04
0.6
volume_frac on=0.05
0.5 0.4 0.3 0.2
0.01 c.)
161
0.1
Reynolds number, Re
Fig. 5. Variation of dimensionless entropy generation j with Reynolds number for ethylene glycoleAl2O3 for different values of a.) Bi b.) volume fraction 4 and c.) nondimensional temperature difference t.
0 0
c.)
1000
2000 Reynolds number
3000
4000
1.2 non-dimensional temperature difference=0.005
Fig. 5(c) renders the variation of j with Re for different values of t. It is seen from this figure that entropy generation rate is higher for higher values of t. This trend is attributed to the fact that a high value of t increases the heat transfer rate to the nanofluid, thereby increasing the temperature gradients inside the fluid and enhancing the entropy generated due to heat transfer. Finally, a comparison of Figs. 5(c) and 2(c) shows that for the same Reynolds number the entropy generated is higher for watereAl2O3 nanofluid compared to ethylene glycoleAl2O3 nanofluid. This corroborates the conclusion of Anand and Nelanti [24]. This is explained by observing that for same Reynolds number, the
Bejan number, Be
1 0.8
non-dimensional temperature difference =0.01
0.6
non-dimensional temperature difference =0.02
0.4 0.2 0 10
510
1010 1510 2010 Reynolds number, Re
2510
Fig. 6. Variation of Bejan number with Reynolds number for ethylene glycoleAl2O3 for different values of a.) Bi b.) volume fraction 4 and c.) non-dimensional temperature difference t.
162
V. Anand / Energy 93 (2015) 154e164
at low values of Bi (Fig. 6(a)), high values of 4 (Fig. 6(b))and low values of t (Fig.6(c)); this trend is attributed to the lower heat transfer irreversibility at these values of Bi, 4 and t. c. Effect on Ns,a: The variation of Ns,a with Reynolds number for different values of Bi, 4 and t is captured in Fig. 7(a)e(c) respectively. It is seen from the figures that Ns,a is above 1 for
a.) 1.16 Bi=1 1.14
Bi=2 Bi=5
1.12
Bi=10
1.1
Ns,a
1.08
1.06 1.04 1.02 1 0.98 0
500
1000
1500
2000
2500
3000
Reynolds number, Re
volume_frac volume_frac volume_frac volume_frac
2
on=0.02 on=0.03 on=0.04 on=0.05
Ns,a
1.5
1
0.5
0
0
c.)
1000
1500 2000 Reynolds number
2500
3000
3500
1.18 non-dimensional temperature difference=0.005
1.16 1.14
non-dimensional temperature difference =0.01
1.12 1.1 Ns,a
500
non-dimensional temperature difference =0.02
1.08 1.06
1.04 1.02 1 0.98 10
510
1010
1510
2010
To summarize, the entropy generated in convection is composed of two different physical mechanisms, namely e entropy generated due to heat transfer and entropy generated due to fluid friction. The entropy generated due to heat transfer (or heat transfer irreversibility) is the dominant mechanism for less viscous fluids and at low Reynolds number because the temperature gradients are higher. On the other hand, entropy generated due to fluid friction is dominant for high viscosity fluids (like ethylene glycol) at higher Reynolds number, since the velocity gradients are higher. Addition of nanoparticles decreases the entropy generated due to heat transfer while increasing the entropy generated due to fluid friction. Thus, from the standpoint of thermodynamics, addition of nanoparticles is recommended only for less viscous fluids, like water, and at low Reynolds number. 4.3. Comparison with other boundary conditions
2.5
b.)
all conditions except at very low Reynolds number. This means that from a strictly thermodynamic/second law point of view, addition of nanoparticles to ethylene glycol is only beneficial at low values of Re. Moreover, it is observed from Fig. 7(b) that addition of nanoparticles reduces the Ns,a at low Re, while it enhances Ns,a at high Re. This can be explained by considering that nanoparticles reduce the entropy generated due to heat transfer while they accentuate the fluid friction irreversibility; at low Re, heat transfer irreversibility dominates whereas fluid friction irreversibility dominates at high Re. The observations of Sections 4.1 and 4.2 have been summarized in Table 4.
2510
Reynolds number, Re Fig. 7. Variation of entropy augmentation number Ns,a with Reynolds number for ethylene glycoleAl2O3 for different values of a.) Bi b.) volume fraction 4 and c.) nondimensional temperature difference t.
As was stated in Section1 of this paper, immersion in an isothermal external fluid is the most generic thermal boundary condition; the other two boundary conditions, namely e UWT (uniform wall temperature) and UHF (uniform heat flux), can be derived as special cases of immersion in an external fluid boundary condition. When the thermal contact between the external fluid and the tube is excellent, the temperature of the wall tends to the uniform temperature of the external fluid. The thermal boundary condition reduces to UWT. This can be seen from Table 2 wherein when Bi / ∞ the Nusselt numbers tend to 3.657, which is also the Nusselt number for laminar convection in circular tube with UWT boundary condition. On the other hand, if the thermal contact between the external fluid and tube wall is very poor, then the wall controls (throttles) the heat flux and the UHF boundary condition is achieved. This can also be seen from Table 2 wherein a value of Bi ¼ 0 leads to Nu ¼ 4.364, which is the Nusselt number for UHF boundary condition for a circular tube. Thus, UWT and UHF serve as bounds for isothermal external fluid boundary condition, as shown by Sparrow and Patankar in [23]. To prove this from the perspective of second law, the nondimensional entropy generation for isothermal external fluid boundary condition is compared with those for UWT and UHF. For the purpose of comparison, the physical system and the inlet and outlet temperatures are kept equal; only the thermal boundary condition is varied. The expressions for entropy generated for UWT and UHF boundary conditions have been taken from Refs. [31] and [32] respectively. The results are plotted in Figs. 8 and 9. Fig. 8 clearly demonstrates that for same inlet and outlet temperatures (and hence the same heat transfer rate), UHF boundary condition generates the least amount of entropy, while UWT generates the highest. The entropy generated for isothermal external fluid is between those generated for UWT and UHF boundary condition. Similar trend is reflected in Fig. 9, which shows the comparison of Bejan numbers for different boundary conditions. This figure
V. Anand / Energy 93 (2015) 154e164
163
Table 4 Summary of the trends of variation of entropy generation with respect to various parameters. Parameter
Entropy generated due to heat transfer
Entropy generated due to fluid friction
Total entropy generated
Re
Decreases
Increases
Bi 4
Increases Decreases
No effect Increases
t
Increases
No effect
For Be > 0.5, For Be < 0.5, Increases For Be > 0.5, For Be < 0.5, Increases
Non-dimensional entropy generaƟon, ψ x 10
100
10
UWT 1 0
1000
2000 3000 Reynolds number
4000
Isothermal External Fluid
0.1
decreases increases decreases increases
assumption valid. Moreover, this study does not capture the effect of variation of tube's physical dimensions e namely length and diameter. A reduction in diameter of the tube, keeping other parameters same, will increase the entropy generated due to fluid friction to a high level e due to corresponding enhancement of velocity gradients. More information on how the physical dimensions of the tube influence the entropy generation rate can be found in Singh et al. [25]. 5. Conclusions
0.01
UHF 0.001
Fig. 8. Comparison of non-dimensional entropy generation for different thermal boundary conditions.
illustrates that the entropy generated due to heat transfer and consequently the Bejan number is the least for UHF and the highest for UWT boundary condition. For isothermal external fluid boundary condition, Bejan number lies between those for UHF and UWT. Finally, the assumptions made in the analysis should also be taken note of. The properties of the base fluid have been taken as constant. This is based on the fact that the temperature rise across the tube is ~1 C, which does not influence the physical properties of the nanofluid to any appreciable extent [8] and thus makes our
In this paper, the entropy generation characteristics of flow of nanofluids in a tube immersed in an isothermal external fluid have been studied. Two different nanofluids, namely e watereAl2O3 and ethylene glycoleAl2O3, have been chosen. The thermal boundary condition employed in this paper e immersion in an isothermal external fluid e is the most generic thermal boundary condition and this underscores the relevance and importance of this study. The effects of external Biot number, volume fraction and nondimensional temperature difference on non-dimensional entropy generation, Bejan number and entropy generation augmentation number have been presented through graphs. The following conclusions can be drawn from this study: 1. Introduction of nanoparticles decreases the entropy generated due to heat transfer whilst increasing entropy generated due to fluid friction. 2. While studying the entropy generation characteristics of nanofluids, Bejan number is a very important parameter. For Be > 0.5,
Fig. 9. Comparison of Bejan number for different thermal boundary conditions. The plots for UWT and isothermal external fluid thermal boundary conditions are shown in the inset.
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V. Anand / Energy 93 (2015) 154e164
increasing the concentration of nanoparticles decreases the total entropy generated; on the other hand, for Be < 0.5, increasing the concentration of nanoparticles will increase the total entropy generation rate. 3. For watereAl2O3, Bejan number is close to 1 and thus addition of nanoparticles decreases the total entropy generation rate. On the other hand, for ethylene glycoleAl2O3 nanofluid, addition of nanoparticles increases the entropy generation rate, except at very low values of Re. This is attributed to the high viscosity of the base fluid e ethylene glycol. 4. The entropy generation for isothermal external fluid is between those for UHF and UWT thermal boundary conditions. The highest entropy is generated for UWT boundary condition while the lowest entropy is generated for UHF boundary condition. This places the UHF boundary condition as the most suitable boundary condition from the perspective of the second law, other parameters remaining the same. Further to the analysis carried out in this paper, research in future may be directed at minimizing the entropy generation rate with respect to the flow rate and the concentration of nanoparticles.
References [1] Choi SUS. Enhancing thermal conductivity of fluids with nanoparticles. ASME Fed 1995;231:99e105. [2] Murshed SMS, Leong KC, Yang C. Thermophysical and electrokinetic properties of nanofluids e a critical review. Appl Therm Eng 2008;28(17e18): 2109e25. [3] Eastman JA, Choi SUS, Li S, Soyez G, Thompson LJ, Di Melfi RJ. Novel thermal properties of nanostructured materials. Mater Sci Forum 1999;312:629e34. [4] Wang XQ, Mujumdar AS. Heat transfer characteristics of nanofluids: a review. Int J Therm Sci 2007;46(1):1e19. [5] Das SK, Choi SUS, Patel HE. Heat transfer in nanofluids e a review. Heat Transf Eng 2006;27(10):3e19. [6] Xuan Y, Li Q. Heat transfer enhancement of nanofluids. Int J Heat Fluid Flow 2000;21:58e64. [7] Lee S, Choi SUS, Li S, Eastman G. Measuring thermal conductivity of fluids containing oxide nanoparticles. J Heat Transf 1999;121:280e9. [8] Das SK, Putra N, Thiesen P, Roetzel W. Temperature dependence of thermal conductivity enhancement for nanofluids. J Heat Transf 2003;125:567e74. [9] Masoumi N, Sohrabi N, Behzadmehr A. A new model for calculating the effective viscosity of nanofluids. J Phys D Appl Phys 2009;42:055501e6.
[10] Corcione M. Empirical correlating equations for predicting the effective thermal conductivity and dynamic viscosity of nanofluids. Energy Convers Manage 2011;52(1):789e93. [11] Khanafer K, Vafai K. A critical synthesis of thermophysical characteristics of nanofluids. Int J Heat Mass Transf 2011;54(19e20):4410e28. [12] Pak BC, Cho YI. Hydrodynamic and heat transfer study of dispersed fluids with submicron metallic oxide particles. Exp Heat Transf 1998;11(2):151e70. [13] Xuan Y, Li Q. Investigation on convective heat transfer and flow features of nanofluids. J Heat Transf 2003;125(1):151e5. [14] Maiga SEB, Nguyen CT, Galanis N, Roy G. Heat transfer behaviors of nanofluids in a uniformly heated tube. J Superlattices Microstruct 2004;35:543e57. [15] Bahiraei M, Hangi M. Numerical simulation of nanofluid application in a C shaped chaotic channel: a potential approach for energy efficiency improvement. Energy 2014:863870. [16] Bahiraei M. Studying nanoparticle distribution in nanofluids considering the effective factors on particle migration and determination of phenomenological constants by Eulerian Lagrangian simulation. Adv Powder Technol 2015;26(3):802810. [17] Bahiraei M, Hangi M. Flow and heat transfer characteristics of magnetic nanofluids: a review. J Magn Magn Mater 2015;374:125e38. [18] Bahiraei M. Effect of particle migration on flow and heat transfer characteristics of magnetic nanoparticle suspensions. J Mol Liq 2015;209:531538. [19] Mahian O, Mahmud S, Heris SZ. Analysis of entropy generation between corotating cylinders using nanofluids. Energy 2012;44(1):438e46. [20] Bianco V, Manca O, Nardini S. Entropy generation analysis of turbulent convection flow of Al2O3ewater nanofluid in a circular tube subjected to constant wall heat flux. Energy Convers Manage 2014;77:306e14. [21] Leong KY, Ong HC. Entropy generation analysis of nanofluid flow in various shapes of cross section ducts. Int Commun Heat Mass Transf 2014;39:72e8. [22] Leong KY, Saidur R, Mahlia TMI, Yau YH. Entropy generation analysis of nanofluid flow in a circular tube subjected to constant wall temperature. Int Commun Heat Mass Transf 2012;39(8):1169e75. [23] Sparrow EM, Patankar S. Relationship among boundary conditions and Nusselt numbers for thermally developed duct flows. J Heat Trans 1977;99: 483e5. [24] Anand V, Nelanti K. Second law analysis of laminar flow in a circular pipe immersed in an isothermal fluid. J Thermodyn 2013. Article ID 234264. [25] Singh PK, Anoop KB, Sundarajan T, Das SK. Entropy generation due to flow and heat transfer in nanofluids. Int J Heat Mass Transf 2010;53:4757e67. [26] Buongiorno J. Convective transport in nanofluids. J Heat Transf 2006;128: 240e50. [27] Xuan Y, Roetzel W. Conceptions for heat transfer correlation of nanofluids. Int J Heat Mass Transf 2000;43:3701e7. [28] Choi SUS, Wang X, Xu W. Thermal conductivity of nano-particle-fluid mixture. J Thermophys Heat Transf 1999;13:474e80. [29] Moghadammi M, Mohammadzade A, Esfehani SAV. Second law analysis of nanofluid flow. Energy Convers Manage 2011;52:1397e405. [30] Incropera FP, DeWitt DP. Fundamentals of heat and mass transfer. 5th ed. Singapore: John Wiley and Sons (Asia); 2006. [31] Sahin AZ. Second law analysis of laminar viscous flow through a duct subjected to constant wall temperature. J Heat Transf 1998;120:76e83. [32] Sahin AZ. Irreversibilities in various duct geometries with constant wall heat flux and laminar flow. Energy 1998;23(6):465e73.
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Entropy generation analysis of laminar flow of a nanofluid in a circular tube immersed in an isothermal external fluid Vishal Anand Cyient Limited (Formerly Infotech Enterprises Limited), Plot No.-2, IT Park, Nanakramguda, Manikonda, Gachiboli, Hyderabad, 500032, India
a r t i c l e i n f o
a b s t r a c t
Article history: Received 26 June 2015 Received in revised form 4 September 2015 Accepted 6 September 2015 Available online xxx
This paper is an analytical study of entropy generation in the laminar flow of nanofluids in a circular tube. The tube is immersed in an isothermal external fluid e which is the most general thermal boundary condition but has not been studied in much detail in literature. Two nanofluids, namely e watereAl2O3 and ethylene glycoleAl2O3 have been chosen for this study. The effects of the external Biot number, nondimensional temperature difference and volume fraction on the entropy generation characteristics of the flow have been shown through graphs and the physical reasoning behind the observed trends has been discussed threadbare. It is shown that the addition of nanoparticles is beneficial only at smaller Reynolds number and for less viscous base fluids. Most importantly, it is proved that the entropy generated in the case of a tube immersed in an isothermal external fluid is bounded by those for uniform heat flux and uniform wall temperature boundary conditions. © 2015 Elsevier Ltd. All rights reserved.
Keywords: Isothermal external fluid Entropy generation Convective heat transfer Bejan number Biot number
1. Introduction Enhancement of heat transfer rate in a heat exchanger tube without changing its physical dimensions is an enduring challenge in thermal sciences and engineering. To meet this challenge, several methods are being researched in industry and in academia. These include and are not limited to- making fins on the internal surfaces of tubes, insertion of twisted tapes, introduction of nanoparticles and others. The word “nanofluid” was coined by Choi [1]. Nanofluid is a suspension of nanoparticles in a base fluid. The principal way in which nanofluids increase the performance of a thermal system is by enhancing the thermal conductivity of the fluid. To that end, a body of work is devoted to investigation of the thermophysical properties of nanofluids as functions of their size and concentration [2e5]. A study by Xuan and Li [6] measured the thermal conductivity of copper based nanofluid. Lee et al. [7] measured the thermal conductivities of oxide nanofluids experimentally. Das et al. [8] included the effect of temperature on the thermal conductivity of water based nanofluids. There have been other models of thermophysical properties presented by researchers like Masoumi et al. [9], Corcione [10] and Khanafer and Vafai [11]. Apart from the modeling of thermophysical properties, there has been considerable research to understand the effect of
E-mail address: [email protected]. http://dx.doi.org/10.1016/j.energy.2015.09.019 0360-5442/© 2015 Elsevier Ltd. All rights reserved.
nanofluids on the heat transfer rate itself. This research has been performed in both experimental and numerical domains. Pak and Cho [12] proposed a correlation for Nusselt number for turbulent convection of nanofluids. Similar correlation was proposed for copper based nanofluids by Xuan and Li [13]. As far as numerical investigation is considered, Maiga et al. [14] studied the laminar and turbulent heat transfer of nanofluids using the commercially available code FLUENT. Bahiraei and Hangi [15] numerically investigated the hydrothermal characteristics of nanofluid in a C shaped chaotic channel using both single-phase and two-phase methods. Bahiraei [16] used the two-phase EulereLagrange method to find the phenomenological constants needed to calculate the diffusion fluxes in nanofluid laminar flow in a circular pipe. A very recent line of research in this domain pertains to magnetic nanofluids e which comprise of non-magnetic base fluid and magnetic nanoparticles. Here, mass transfer and heat transfer can be controlled by applying magnetic fields. Some of the relevant research work in this area can be accessed in Refs. [17,18]. A prominent disadvantage concerning the use of nanofluids is that due to the concomitant increase in viscosity, the pressure drop (and thereby, the pumping power required) is higher. This means that the heat transfer engineer/researcher should continuously -vis the evaluate the advantage of high heat transfer rate vis-a penalty of high pumping power requirement. An effective tool for this evaluation is the entropy generation analysis. Entropy generated due to heat transfer decreases while the entropy generated due to pressure difference increases with introduction of
V. Anand / Energy 93 (2015) 154e164
Nomenclature A Be Bi Cp D Ec f FF h b h he k L m_ Ns,a Nu b Nu P Pr q_ 00 _ q0 Q_ r Re s S_ St
cross-sectional area (m2) Bejan number Biot number he r=k specific heat capacity (J/kg K) cross sectional diameter (m) Eckert number (defined in Eq. (30)) friction factor entropy generated due to fluid friction internal heat transfer coefficientq_ 00 =ðTw TÞ (W/m2K) overall heat transfer coefficient q_ 00 =ðTe TÞ (W/m2K) external heat transfer coefficient q_ 00 =ðTe Tw Þ (W/m2K) thermal conductivity (W/m K) duct length (m) mass flow rate (kg/s) entropy generation augmentation number internal Nusselt number hD/k overall Nusselt number b hD/k pressure (Pa) Prandtl number local heat flux (W/m2) heat rate per unit length (W/m) heat rate (W) radius of tube (m) Reynolds number specific entropy (J/kg K)
T U
155
temperature (K) average fluid velocity (m/s)
Greek letters a dimensionless parameter (defined in Eq. (28)) t dimensionless temperature difference (defined in Eq. (29)) l dimensionless length (defined in Eq. (15)) lL dimensionless length (defined in Eq. (17)) j dimensionless entropy generation rate (defined in Eq. (31)) 4 volume fraction of nanoparticles DP pressure drop (Pa) r fluid density (kg/m3) m fluid viscosity (Ns/m2) p1 dimensionless parameter (defined in Eq. (33)) p2 dimensionless parameter (defined in Eq. (34)) Subscripts avg average bf base fluid e external i internal nf nanofluid np nanoparticle out outlet ref reference w wall
entropy rate (W/K) Stanton number (defined in Eq. (13))
nanoparticles in the fluid. Thus, it is possible e indeed desirable, to gauge the effectiveness of nanofluids by studying their total entropy generation rates. The entropy generation of nanofluids between two co-rotating cylinders was investigated by Mahian et al. [19]. They showed that it is possible to minimize the entropy generated in the nanofluid with respect to the volume fraction of the nanoparticles. Bianco et al. [20] studied the entropy generated in nanofluid under turbulent conditions in a circular tube subject to constant wall heat flux. They showed that the optimum nanoparticle concentration for minimum entropy generation is lower at higher Reynolds number. Leong et al. [21] studied the effect of cross-sectional shape on the entropy generation analyses of nanofluids. They found out that the circular tube generated the least amount of entropy compared to the tubes of other crosssections. The same research group also studied the entropy generation in a tube with uniform wall temperature for nanofluids derived from alumina and titanium dioxide [22]. They concluded that titanium oxide nanofluids generate less entropy than that generated by alumina based nanofluids. It is to be noted that all the cases of entropy generation analysis considered in extant literature have been based on one of the two common thermal boundary conditions: UWT (uniform wall temperature) and UHF (uniform heat flux). Sparrow and Patankar in a seminal study in 1977 [23] showed that these two boundary conditions are special cases of a more generic boundary condition, namely e immersion in an isothermal external fluid. When the thermal contact between the isothermal external fluid and the tube wall is superior, UWT boundary condition is obtained. On the other hand, if the external thermal contact is poor, the boundary condition reduces to UHF. The entropy generation analysis for laminar flow in a circular tube immersed in an isothermal external fluid was performed by
the author in an earlier publication [24], but that analysis was only for a single phase fluid. The concept of heat transfer enhancement by the introduction of nanoparticles was outside the scope of that paper e but is included in the current paper. This paper deals with entropy generation analyses of nanofluid flow in a circular tube immersed in an isothermal external fluid. Two nanofluids: watereAl2O3 and ethylene glycoleAl2O3 have been chosen for the study. These two nanofluids have been chosen primarily because they are the most commonly used nanofluids in industry and in research. Moreover, the fact that the difference in these two nanofluids is only due to the difference in base fluid e the nanoparticles are the same e makes the assessment more systematic and organized. The novelty of this paper derives from the choice of the thermal boundary condition, namely e immersion in an isothermal external fluid. This is the most common thermal boundary condition seen in nature as well as in industry. For example, the thermal characteristics of a pipe carrying water in ambient surroundings, an oil pipeline in the ocean-bed, and heat exchanger tubes in shell-and-tube heat exchangers will be governed by this thermal boundary condition. It is shown in the results section that both UHF and UWT boundary conditions serve as lower and upper bounds respectively for the isothermal external fluid boundary condition and are thus only special cases of a more generic boundary condition. 2. Problem statement The schematic of the physical system under consideration is shown in Fig. 1. It consists of a circular tube immersed in an isothermal external fluid, which is at uniform and constant temperature Te. The diameter of the tube is D. The nanofluid (watereAl2O3 or ethylene glycoleAl2O3) enters the tube at temperature Ti.
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V. Anand / Energy 93 (2015) 154e164
These equations were proposed by Maiga et al. [14] and are derived from the experimental works carried out by Lee et al. [7] and Choi et al. [28]. Other researchers in nanofluid research have also used the same correlations for watereAl2O3 and ethylene glycoleAl2O3 nanofluids [29].
3.2. Entropy generation analysis
Fig. 1. The schematic of the physical system under study. Also shown is an infinitesimal control volume.
The flow is steady, laminar and hydrodynamically and thermally fully developed. The thermophysical properties of the base fluids and the nanoparticles have been taken as constants.
3.1. Thermophysical properties of nanofluids
(1)
Here np refers to nanoparticle, bf refers to base fluid, nf refers to nanofluid, while 4 refers to the volume fraction of nanoparticles in the fluid. The specific heat of the nanofluid is evaluated by using the following formula:
Cp;nf
2
mnf ¼ 1234 þ 7:34 þ 1 mbf knf
¼ 4:9742 þ 2:724 þ 1 kbf
(2)
(3) (4)
For ethylene glycoleAl2O3
2
(8)
where b h is the overall heat transfer coefficient, between the external fluid and the internal fluid. Solving the two equations above for T(x), we obtain:
TðxÞ ¼ Te ðTe Ti Þe
b
4 h x rUDCp
(9)
Here, U is the average velocity of the flow. The above equation can be written in terms of non-dimensional parameters as:
TðxÞ ¼ Te ðTe Ti Þe
_
Nu 4Re:Pr l
¼ Te ðTe Ti Þeð4StlÞ
(10)
where, Re is the Reynolds number based on tube diameter:
This formula is based on the assumption that the nanoparticle and base fluid are at thermal equilibrium; there is no heat exchange between a nanoparticle and its surrounding base fluid and they are at same temperature. This formula was presented first by Buongiorno [26] and has been validated experimentally by Pak and Cho [12] and Xuan and Roetzel [27]. The viscosity and thermal conductivity of the nanofluids are calculated using the following formulae: For watereAl2O3:
(7)
b q_ 0 ¼ hpDðT e TÞ
Any theoretical analysis of nanofluids behavior leans heavily on the models used to determine their thermophysical properties as a function of constituents' properties and concentration. For an understanding of how modeling of properties of nanofluids affects their entropy generation analyses, the reader is directed to Singh et al. [25]. The density of the nanofluid is evaluated by using the following formula which is commonly employed for mixtures:
ð1 4Þrbf Cp;bf þ 4rnp Cp;np ¼ rnf
_ p dT ¼ q_ 0 dx mC
Here, Cp is the specific heat of the fluid and m_ is the mass flow rate. q_ 0 is the rate of heat transfer per unit length from the external fluid to the internal fluid and is given by
3. Mathematical formulation
rnf ¼ ð1 4Þrbf þ 4rnp
The entropy generation analysis of laminar flow in a tube immersed in an isothermal, external fluid has been carried out by this author in an earlier publication [24] and is presented here in brief: Consider a CV (control volume) of infinitesimal length dx, in the flow. The energy balance for this CV, (neglecting axial conduction), is given by:
mnf ¼ 3064 0:194 þ 1 mbf
(5)
knf ¼ 28:90542 þ 2:82734 þ 1 kbf
(6)
Re ¼
rUD m
(11)
b is the overall Nusselt number, given as: Here, Nu
b b ¼ hD Nu k
(12)
St is the Stanton number, given by:
St ¼
b b h Nu ¼ rUCp Re:Pr
(13)
r is the fluid density.Pr is the Prandtl number:
Pr ¼
mCp k
(14)
m is the fluid viscosity and k is the thermal conductivity of the fluid. And l is the non-dimensional duct length, given by:
l¼
x D
The temperature at the outlet of the duct Tout is given by
(15)
V. Anand / Energy 93 (2015) 154e164
Tout ¼ TðLÞ ¼ Te ðTe Ti Þeð4StlL Þ
(16)
where,
L lL ¼ D
Tout ¼ TðLÞ ¼ Te ðTe Ti Þe
b
a¼1
N ukpL _ p mC
(18)
The rate of total heat transferred to this CV is given by
_ p ðTout Ti Þ Q_ ¼ mC _ p Te Ti ðTe Ti Þe4StlL ¼ mC
(19)
dS_gen ¼
q_ 0 dx _ þ mds Tw
Here Tw is the temperature of the wall and ds is the infinitesimal entropy change across this CV. Now,
dS_gen ¼
b e TÞpDdx hðT _ þ mds b hðTe TÞ þ T h
(22)
For an incompressible fluid:
_ ¼ m_ mds
Cp dT dP m_ rT T
(23)
dP ¼ dx 2D
_ p dT mfU _ 2 mC dx þ 2DT T
(25)
Substituting in the Eq. (22) we get:
b hðTe TÞpDdx dT fU 2 _ p þ m_ þ mC dx ¼ b T 2DT hðTeTÞ þ T h
j¼
1 te4StlL fEct e4StlL t 1 1 tae4StlL þ ln ln þ ln a 8St 1t 1 ta 1t (31)
The significance of dimensionless entropy generation is that it has been non-dimensionalized by using mass flow rate (and specific heat). This allows the researcher to make a valid comparison of the non-dimensional entropy generation of flows with different mass flow rates. The Eq. (31) can also be written as function of Re in place of St ! b Nu St ¼ RePr
1 tep1 =Re 1 1 taep1 =Re ln þ ln a 1 ta 1t p =Re e 1 t þ p2 Re2 ln 1t
Substitute the value of T(x) and integrate from x ¼ 0 to x ¼ L, we obtain the total entropy generated in the duct as:
(32)
where
p1 ¼
b 4 Nul Lk mCp
(33)
p2 ¼
ðf ReÞm3 b e ðDrÞ2 k 8 NuT
(34)
Bejan number: Bejan number is defined as:
Be ¼ (26)
(30)
S_gen _ p mC
(24)
Thus we get:
(29)
In literature, dimensionless entropy generation has been used to quantify the entropy generated. It is given by:
j¼
where the pressure drop per unit length is related to friction factor ( f ) as:
f rU 2
U2 Cp ðTe Ti Þ
(21)
Substitute the values of q_ 0 and Tw from Eq. (21) into Eq. (20), the following expression for infinitesimal entropy generation rate is obtained:
(28)
Ec is the Eckert number:
¼
b q_ 0 ¼ hpDðT e TÞ ¼ hpDðTw TÞ b hðTe TÞ þT 0Tw ¼ h
Te Ti Te
Ec ¼ (20)
b Nu Nu
where, Nu is the internal Nusselt number ¼ hD/k.t is the nondimensional temperature difference, defined as:
t¼
The entropy generated in this infinitesimal CV is given by:
dS_gen
(27)
(17)
Writing Eq. (16) in terms of mass flow rate, we obtain:
_ ¼ mds
" 1 te4StlL 1 1 tae4StlL _ p ln þ ln S_gen ¼ mC 1t a 1 ta # fEct e4StlL t þ ln 8St 1 t
157
EntropyGeneratedDueToHeatTransfer TotalEntropyGenerated
At Be ¼ 1, all the entropy generated is due to heat transfer. At Be ¼ 0, all the entropy generated is due to fluid friction. In the current analysis, the entropy generated due to fluid friction is given by:
158
V. Anand / Energy 93 (2015) 154e164
FF ¼ lim j ¼ p1 p2 Re Te /Ti
(35)
where FF ¼ Entropy generated due to fluid friction only. So,
Be ¼
j FF FF ¼1 j j
(36)
Entropy augmentation number: To measure the efficacy of any heat transfer augmentation technique, its performance must be compared with respect to the original system. Since the current analysis deals with entropy generation, the relevant augmentation number is the entropy augmentation number, defined as:
Ns;a ¼
Entropy Generated in System with nanofluids Entropy Generated in System with basefluids only
Table 2 b as a function of Bi. Nu and Nu Bi
Nu
b Nu
0 0.1 0.25 0.5 1 2 5 10 100 ∞
4.364 4.330 4.284 4.221 4.122 3.997 3.840 3.758 3.663 3.657
0 0.1909 0.4471 0.8075 1.345 1.998 2.773 3.163 3.597 3.657
Taken from Ref. [23].
Table 3 Different parameters used in the analysis.
(37) By this definition, a values of Ns,a lower than 1 is desirable. Only then, the introduction of nanoparticles is a thermodynamically viable option. In this analysis based on the above equations, the properties of the base fluids are taken from Incropera et al. [30] while those of Al2O3 have been taken from Ref. [21]. They have been presented in Table 1. For different values of volume concentration of Al2O3, these properties are used to calculate the thermophysical properties of the nanofluid using the equations presented in Section 3. b as functions of external Biot number Bi The values of Nu and Nu have been taken from Ref. [23] and presented in Table 2. The value of friction factor for laminar flow in a tube is f ¼ 64/Re from Ref. [30]. Similarly, the numerical values of the other parameters used in the analysis have been tabulated in Table 3. These values are commensurate with those used elsewhere in archival literature. 4. Results The problem statement has been presented in Section 2. Pursuant to the theme of this study, the constitutive equations for the thermophysical properties of nanofluids have been presented in Section 3. The expressions for the non-dimensional entropy generation rate, Bejan number and non-dimensional entropy augmentation number have been derived in the later part of Section 3. Now, in this section, we proceed to present the results and analyze the trends observed therein. In this section, the results have been presented for both water and ethylene glycol. A comparison has also been made between the results of isothermal external fluid boundary condition presented here and those for the more common thermal boundary conditions of uniform wall temperature and uniform heat flux. 4.1. WatereAl2O3 a. Effect on j: For the nanofluid combination of watereAl2O3, the variation of non-dimensional entropy generation rate j with Re is shown in Fig. 2. The effect of Biot number is captured in Fig. 2(a), while Fig. 2(b) and (c) show the effects of the volume
Parameters
Numerical values
Te D L
310 K 0.08 m 1.0 m
concentration 4 and non-dimensional temperature difference t respectively. From Fig. 2, it is observed that the non-dimensional entropy generation rate j decreases with increase in Re. The reason for this trend is attributed to the fact that as the Reynolds number increases, the temperature at any point in the tube tends to the inlet temperature ðfrom Eq: 10 lim TðxÞ ¼ Ti Þ, an increase in Reynolds Re/∞ number causes the temperature gradients inside the fluid to decrease; the whole fluid control volume tends to a uniform temperature. As a result of the decrease in temperature gradients, the entropy generated inside the fluid due to heat transfer decreases. On the other hand, an increase in Reynolds number causes the entropy generated due to fluid friction to increase. But, as shown later in Section 4.1.b, the Bejan number for watereAl2O3 nanofluid is close to 1. This indicates that the entropy generated due to heat transfer is much greater than the entropy generated due to fluid friction and thus the overall entropy generation rate is influenced primarily by the trends in entropy generation due to heat transfer. Fig. 2(a) shows that the non-dimensional entropy generation rate j increases with increase in Biot number Bi. This trend is explained by appreciating that an increase in Bi means that the thermal resistance between the isothermal external fluid and the wall of the tube decreases. The heat transfer rate to the nanofluid is thus enhanced and temperature gradients inside the fluid increases. This translates into higher entropy generation rate. Fig. 2(b) shows that the entropy generation rate decreased with rise in volume fraction 4. To understand this trend, we recognize that the motive of adding nanoparticles is to augment the heat transfer rate by improving the thermal conductivity of the nanofluid. The increase in thermal conductivity leads to decrease in temperature gradients inside the fluid which reduces the entropy generated due to heat transfer. On the other hand, increase in volume fraction 4 also leads to increase in viscosity of the nanofluid
Table 1 Properties of the base fluid and nanoparticles.
Water Ethylene glycol Al2O3
Thermal conductivity (W/m K)
Thermal specific Heat (J/kg K)
Viscosity (Ns/m2)
Density (kg/m3)
0.613 0.252 40
4180 2415 880
0.000855 0.0157
1000 1114.4 3900
V. Anand / Energy 93 (2015) 154e164
a.)
a.)
10
1 0.9999 0.9998 0.9997 0.9996 0.9995 0.9994 0.9993 0.9992 0.9991
Bi=1.0
8
Bi=2.0
6
Bi=5.0
4
Bi=10.0
Bejan number, Be
Non-dimensional entropy 6 generation, ψ x10
12
159
2 0
Bi=1.0 Bi=2.0 Bi=5.0 Bi=10.0
0
0
2000
4000
1000
2000
3000
4000
5000
Reynolds number, Re
6000
Reynolds number, Re 1
b.)
b.)
0.9999 volume_frac on=0.01
3.5
volume_frac on=0.02
volume_frac on=0.03
3
volume_frac on=0.04
2.5
volume_frac on=0.05
volume_frac on=0.01
0.9996
volume_frac on=0.02
0.9995
volume_frac on=0.03
0.9994
volume_frac on=0.04
1.5
0.9993
volume_frac on=0.05
1
0.9992 500
0.5
1500
2500
3500
4500
Reynolds number, Re
0 1500
2500 3500 Reynolds number, Re
4500
5500
16 Non-dimensional entropy generation, ψ x10
0.9997
2
500 c.)
0.9998 Bejan number, Be
4
14
Non-dimensional temperature difference=0.005
12
Non-dimensional temperature difference =0.01
10 8
Non-dimensional temperature difference =0.02
6
c.)
1.0002 1
Bejan number, Be
Dimensionless Entropy Generation, ψ x10
4.5
0.9998
Non-dimensional temperature difference=0.005 Non-dimensional temperature difference =0.01 Non-dimensional temperature difference =0.02
0.9996 0.9994 0.9992 0.999 0.9988 0.9986 0
1000
2000
3000
4000
5000
Reynolds number, Re
4
Fig. 3. Variation of Bejan number with Reynolds number for watereAl2O3 for different values of a.) Bi b.) volume fraction 4 and c.) non-dimensional temperature difference t.
2 0 0
2000
4000
6000
Reynolds number, Re Fig. 2. Variation of dimensionless entropy generation j with Reynolds number for watereAl2O3 for different values of a.) Bi b.) volume fraction 4 and c.) non-dimensional temperature difference t.
and a corresponding increase in entropy generated due to fluid friction. But as mentioned earlier, for this nanofluid e watereAl2O3, the entropy generated due to fluid friction is miniscule, so the total entropy generation rate follows the trend of entropy generated due to heat transfer. And thus, entropy generation rate decreases with addition of nanoparticle. Fig. 2(c) shows that the entropy generation increases with increase in non-dimensional temperature difference on account of increase in heat transfer rate and the consequent increase in temperature gradients inside the fluid. Effect on Be: Fig. 3 shows the variation of Be with Re for watereAl2O3 nanofluid. The effects of Bi, 4 and t on Bejan number have been captured in Fig. 3(a)e(c) respectively. The first observation is that for this nanofluid, Bejan
number is close to 1 for all the values of parameters considered. This leads to the conclusion that for a mildly viscous base fluid like water, the entropy generated due to heat transfer is overwhelmingly high, irrespective of the volume fraction of nanoparticle. We also observe that Bejan number decreases with increase in Re. This is attributed to the twin reasons of increase in entropy generated due to fluid friction at high Re, and decrease in entropy generated due to heat transfer because of decline in temperature gradients inside the fluid. It is seen from Fig. 3(a) that Be increases with increase in Bi. This trend is attributed to the increased heat transfer rate to the nanofluid at higher Bi, which increases the temperature gradients inside the fluid and consequently the entropy generated due to heat transfer increases. Fig. 3(b) shows that an increase in volume fraction 4 leads to a decrease in value of Bejan number. To explain this trend, we recognize that an increase in volume fraction increases the thermal conductivity of the nanofluid. This brings down the temperature gradients inside the fluid and thereby reduces the entropy generated due to heat transfer. Further to this, the entropy generated due
V. Anand / Energy 93 (2015) 154e164
to fluid friction also increases because of increase in viscosity. These two phenomena work in tandem to bring down the value of Bejan number. Fig. 3(c) shows that Be increases with increase in t on account of enhanced heat transfer.
a.)
b. Effect on Ns,a: As mentioned earlier, an important parameter to measure the performance of any heat transfer augmentation technique is the augmentation number which quantifies the -vis that of thermal performance of the augmented system vis-a the original system. Since this study deals with second law analysis of nanofluids, the relevant augmentation number is the entropy augmentation number, defined as the ratio of entropy generated in nanofluids over the entropy generated in base fluids (see Eq. (37)). By this definition, a value of Ns,a lower than 1 is more desirable.
4.2. Ethylene glycoleAl2O3 Next we consider another nanofluid e a combination of ethylene glycol and Al2O3. As seen from Table 1, the viscosity of ethylene glycol is about 400 times that of water. In this section, we will understand how this disparity in viscosity (between water and ethylene glycol) affects the entropy generation characteristics of ethylene glycoleAl2O3 nanofluid. a. Effect on j: The variation of j with Reynolds number for ethylene glycoleAl2O3 nanofluid is shown in Fig. 5. The effects of Bi, 4 and t on j are rendered through Fig. 5(a)e(c) respectively. From Fig. 5(a), it is seen that the value of j first decreases and then increases with increase in Re- especially at low values of Bi (Bi ¼ 1,2). This is explained as follows. Total entropy generation rate is the sum of entropy generation rate due to heat transfer and that due to fluid friction. Initially, j decreases with increase in Re, because an increase in Re decreases the temperature gradients inside the fluid, thereby decreasing the heat transfer irreversibility. But at lager Re, due to the high viscosity of ethylene glycol, the entropy generated due to fluid friction increases and compensates for the decline in entropy generated due to heat transfer. This
0.946
Bi=1.0
0.945
Bi=2.0
0.944
Bi=5.0
0.943
Bi=10.0
0.942 0.941 0.94 0.939 0
2000
4000
6000
Reynolds number, Re
Ns,a
b.)
0.99 0.97 0.95 0.93 0.91 0.89 0.87 0.85 0.83 0.81 0.79
volume_frac on=0.01 volume_frac on=0.02 volume_frac on=0.03 volume_frac on=0.05 voulme_frac on=0.04
0
2000
4000
6000
Reynolds number, Re
c.)
0.945
Non-dimensional temperature difference=0.005
0.944
Non-dimensional temperature difference =0.01
0.943 Ns,a
Fig. 4 depicts the variation of Ns,a with Reynolds number for the watereAl2O3 nanofluid. The effects of Bi, 4 and t on Ns,a have been shown in Fig. 4(a)e(c) respectively. The foremost observation from these figures is that Ns,a is less than 1 for all values of Re, Bi, t and 4. Thus, from the standpoint of the second law, addition of nanoparticles like Al2O3 to water always improves the thermodynamic performance of the heat exchange tube. We also notice that the value of Ns,a keeps decreasing with increase in Re. This is attributed to the fact that an increase in Re causes the entropy generated due to heat transfer to decrease and this decrease in entropy generation is accentuated by the introduction of the nanoparticles. From Fig. 4(a), we observe that Ns,a is the least for smallest Bi, all other parameters kept same. This is explained as follows. For Bi ¼ 1, the heat transfer rate to the fluid is the least and the temperature field inside the fluid is the weakest. The introduction of nanoparticles is thus able to make a stronger impact on the temperature field inside the fluid, reducing the temperature gradients by improving the thermal conductivity. This in turn reduces the entropy generation rate by a higher amount. Fig. 4(b) shows that Ns,a decreases with increase in volume fraction 4. This trend is also along expected lines as introduction of higher concentration of Al2O3 leads to steeper decline in entropy generated due to heat transfer by improving thermal conduction inside the fluid.
0.947
Ns,a
160
0.942
Non-dimensional temperature difference =0.02
0.941 0.94 0.939 0
2000
4000
6000
Reynolds number, Re Fig. 4. Variation of entropy augmentation number Ns,a with Reynolds number for watereAl2O3 for different values of a.) Bi b.) volume fraction 4 and c.) non-dimensional temperature difference t.
results in net increase in total entropy generation rate. This trend is more apparent at lower values of Bi, because at lower values of Bi, the entropy generation due to heat transfer is even lower. Fig. 5(b) shows the variation of j with Re for different values of 4. It can be discerned from the figure that at low values of Reynolds number, j is higher for smaller 4, while at higher values of Reynolds number, j is higher for larger 4. This can be explained as follows. At low Re, entropy generated due to heat transfer is higher as opposed to the situation at high Re e where entropy generated due to fluid friction dominates. Entropy generated due to heat transfer decreases with increase in 4, while entropy generated due to fluid friction increases with 4. This accounts for decrease in total entropy generation rate with 4 at lower Re and the corresponding increase at higher Re.
V. Anand / Energy 93 (2015) 154e164
velocity of water will be less than that of ethylene glycol (due to disparity in value of viscosity). So the temperature gradients inside water are higher and thus entropy generated due to heat transfer is also high. This leads to total (dimensionless) entropy generation being higher for water than that for ethylene glycol.
Non dimensional entropy genera on, ψ x 106
100
10
Bi=1
Bi=2
Bi=5
Bi=10
b. Effect on Be: The variation of Be with Reynolds number is rendered in Fig. 6. Fig. 6(a) captures the effect of Bi, while the effects of 4 and those of t are presented in Fig. 6(b) and (c) respectively.
1 0
1000 2000 Reynolds number, Re
3000
0.1
It is seen from the figure that Be decreases with increase in Reynolds number. This is along expected lines as an increase in Re leads to decrease in heat transfer irreversibility while at the same time it accentuates the fluid friction irreversibility. The most important observation is that Bejan number falls below 0.5 for high value of Re for certain values of Bi, 4 and t. This is quite contrary to the trend observed in water, where Be remained close to 1. This happens because ethylene glycol is much more viscous than water, so the entropy generated due to fluid friction is very high at higher Re. The lowering of Bejan number below 0.5 is more likely to occur
0.01 a.)
10
volume_frac on=0.02
Non-dimensional entropy genera on, ψ x 106
volume_frac on=0.03 volume_frac on=0.04 volume_frac on=0.05
1
500
1000
1500
2000
2500
3000
3500
a.)
1.2
1 Bejan number, Be
0
0.1
b.)
0.01
Bi=1
Bi=2
Bi=5
Bi=10
0.8 0.6 0.4 0.2 0 0
non-dimensional temperature difference=0.005
500
1000
1500
2000
2500
3000
Reynolds number, Re
non-dimensional temperature difference =0.01
10
1
b.) non-dimensional temperature difference =0.02
volume_frac on=0.02
0.9
volume_frac on=0.03
0.8
1
10
510
1010
1510
2010
2510
Bejan number, Be
Non-dimensional entropy genera on, ψ x106
100
0.1
0.7
volume_frac on=0.04
0.6
volume_frac on=0.05
0.5 0.4 0.3 0.2
0.01 c.)
161
0.1
Reynolds number, Re
Fig. 5. Variation of dimensionless entropy generation j with Reynolds number for ethylene glycoleAl2O3 for different values of a.) Bi b.) volume fraction 4 and c.) nondimensional temperature difference t.
0 0
c.)
1000
2000 Reynolds number
3000
4000
1.2 non-dimensional temperature difference=0.005
Fig. 5(c) renders the variation of j with Re for different values of t. It is seen from this figure that entropy generation rate is higher for higher values of t. This trend is attributed to the fact that a high value of t increases the heat transfer rate to the nanofluid, thereby increasing the temperature gradients inside the fluid and enhancing the entropy generated due to heat transfer. Finally, a comparison of Figs. 5(c) and 2(c) shows that for the same Reynolds number the entropy generated is higher for watereAl2O3 nanofluid compared to ethylene glycoleAl2O3 nanofluid. This corroborates the conclusion of Anand and Nelanti [24]. This is explained by observing that for same Reynolds number, the
Bejan number, Be
1 0.8
non-dimensional temperature difference =0.01
0.6
non-dimensional temperature difference =0.02
0.4 0.2 0 10
510
1010 1510 2010 Reynolds number, Re
2510
Fig. 6. Variation of Bejan number with Reynolds number for ethylene glycoleAl2O3 for different values of a.) Bi b.) volume fraction 4 and c.) non-dimensional temperature difference t.
162
V. Anand / Energy 93 (2015) 154e164
at low values of Bi (Fig. 6(a)), high values of 4 (Fig. 6(b))and low values of t (Fig.6(c)); this trend is attributed to the lower heat transfer irreversibility at these values of Bi, 4 and t. c. Effect on Ns,a: The variation of Ns,a with Reynolds number for different values of Bi, 4 and t is captured in Fig. 7(a)e(c) respectively. It is seen from the figures that Ns,a is above 1 for
a.) 1.16 Bi=1 1.14
Bi=2 Bi=5
1.12
Bi=10
1.1
Ns,a
1.08
1.06 1.04 1.02 1 0.98 0
500
1000
1500
2000
2500
3000
Reynolds number, Re
volume_frac volume_frac volume_frac volume_frac
2
on=0.02 on=0.03 on=0.04 on=0.05
Ns,a
1.5
1
0.5
0
0
c.)
1000
1500 2000 Reynolds number
2500
3000
3500
1.18 non-dimensional temperature difference=0.005
1.16 1.14
non-dimensional temperature difference =0.01
1.12 1.1 Ns,a
500
non-dimensional temperature difference =0.02
1.08 1.06
1.04 1.02 1 0.98 10
510
1010
1510
2010
To summarize, the entropy generated in convection is composed of two different physical mechanisms, namely e entropy generated due to heat transfer and entropy generated due to fluid friction. The entropy generated due to heat transfer (or heat transfer irreversibility) is the dominant mechanism for less viscous fluids and at low Reynolds number because the temperature gradients are higher. On the other hand, entropy generated due to fluid friction is dominant for high viscosity fluids (like ethylene glycol) at higher Reynolds number, since the velocity gradients are higher. Addition of nanoparticles decreases the entropy generated due to heat transfer while increasing the entropy generated due to fluid friction. Thus, from the standpoint of thermodynamics, addition of nanoparticles is recommended only for less viscous fluids, like water, and at low Reynolds number. 4.3. Comparison with other boundary conditions
2.5
b.)
all conditions except at very low Reynolds number. This means that from a strictly thermodynamic/second law point of view, addition of nanoparticles to ethylene glycol is only beneficial at low values of Re. Moreover, it is observed from Fig. 7(b) that addition of nanoparticles reduces the Ns,a at low Re, while it enhances Ns,a at high Re. This can be explained by considering that nanoparticles reduce the entropy generated due to heat transfer while they accentuate the fluid friction irreversibility; at low Re, heat transfer irreversibility dominates whereas fluid friction irreversibility dominates at high Re. The observations of Sections 4.1 and 4.2 have been summarized in Table 4.
2510
Reynolds number, Re Fig. 7. Variation of entropy augmentation number Ns,a with Reynolds number for ethylene glycoleAl2O3 for different values of a.) Bi b.) volume fraction 4 and c.) nondimensional temperature difference t.
As was stated in Section1 of this paper, immersion in an isothermal external fluid is the most generic thermal boundary condition; the other two boundary conditions, namely e UWT (uniform wall temperature) and UHF (uniform heat flux), can be derived as special cases of immersion in an external fluid boundary condition. When the thermal contact between the external fluid and the tube is excellent, the temperature of the wall tends to the uniform temperature of the external fluid. The thermal boundary condition reduces to UWT. This can be seen from Table 2 wherein when Bi / ∞ the Nusselt numbers tend to 3.657, which is also the Nusselt number for laminar convection in circular tube with UWT boundary condition. On the other hand, if the thermal contact between the external fluid and tube wall is very poor, then the wall controls (throttles) the heat flux and the UHF boundary condition is achieved. This can also be seen from Table 2 wherein a value of Bi ¼ 0 leads to Nu ¼ 4.364, which is the Nusselt number for UHF boundary condition for a circular tube. Thus, UWT and UHF serve as bounds for isothermal external fluid boundary condition, as shown by Sparrow and Patankar in [23]. To prove this from the perspective of second law, the nondimensional entropy generation for isothermal external fluid boundary condition is compared with those for UWT and UHF. For the purpose of comparison, the physical system and the inlet and outlet temperatures are kept equal; only the thermal boundary condition is varied. The expressions for entropy generated for UWT and UHF boundary conditions have been taken from Refs. [31] and [32] respectively. The results are plotted in Figs. 8 and 9. Fig. 8 clearly demonstrates that for same inlet and outlet temperatures (and hence the same heat transfer rate), UHF boundary condition generates the least amount of entropy, while UWT generates the highest. The entropy generated for isothermal external fluid is between those generated for UWT and UHF boundary condition. Similar trend is reflected in Fig. 9, which shows the comparison of Bejan numbers for different boundary conditions. This figure
V. Anand / Energy 93 (2015) 154e164
163
Table 4 Summary of the trends of variation of entropy generation with respect to various parameters. Parameter
Entropy generated due to heat transfer
Entropy generated due to fluid friction
Total entropy generated
Re
Decreases
Increases
Bi 4
Increases Decreases
No effect Increases
t
Increases
No effect
For Be > 0.5, For Be < 0.5, Increases For Be > 0.5, For Be < 0.5, Increases
Non-dimensional entropy generaƟon, ψ x 10
100
10
UWT 1 0
1000
2000 3000 Reynolds number
4000
Isothermal External Fluid
0.1
decreases increases decreases increases
assumption valid. Moreover, this study does not capture the effect of variation of tube's physical dimensions e namely length and diameter. A reduction in diameter of the tube, keeping other parameters same, will increase the entropy generated due to fluid friction to a high level e due to corresponding enhancement of velocity gradients. More information on how the physical dimensions of the tube influence the entropy generation rate can be found in Singh et al. [25]. 5. Conclusions
0.01
UHF 0.001
Fig. 8. Comparison of non-dimensional entropy generation for different thermal boundary conditions.
illustrates that the entropy generated due to heat transfer and consequently the Bejan number is the least for UHF and the highest for UWT boundary condition. For isothermal external fluid boundary condition, Bejan number lies between those for UHF and UWT. Finally, the assumptions made in the analysis should also be taken note of. The properties of the base fluid have been taken as constant. This is based on the fact that the temperature rise across the tube is ~1 C, which does not influence the physical properties of the nanofluid to any appreciable extent [8] and thus makes our
In this paper, the entropy generation characteristics of flow of nanofluids in a tube immersed in an isothermal external fluid have been studied. Two different nanofluids, namely e watereAl2O3 and ethylene glycoleAl2O3, have been chosen. The thermal boundary condition employed in this paper e immersion in an isothermal external fluid e is the most generic thermal boundary condition and this underscores the relevance and importance of this study. The effects of external Biot number, volume fraction and nondimensional temperature difference on non-dimensional entropy generation, Bejan number and entropy generation augmentation number have been presented through graphs. The following conclusions can be drawn from this study: 1. Introduction of nanoparticles decreases the entropy generated due to heat transfer whilst increasing entropy generated due to fluid friction. 2. While studying the entropy generation characteristics of nanofluids, Bejan number is a very important parameter. For Be > 0.5,
Fig. 9. Comparison of Bejan number for different thermal boundary conditions. The plots for UWT and isothermal external fluid thermal boundary conditions are shown in the inset.
164
V. Anand / Energy 93 (2015) 154e164
increasing the concentration of nanoparticles decreases the total entropy generated; on the other hand, for Be < 0.5, increasing the concentration of nanoparticles will increase the total entropy generation rate. 3. For watereAl2O3, Bejan number is close to 1 and thus addition of nanoparticles decreases the total entropy generation rate. On the other hand, for ethylene glycoleAl2O3 nanofluid, addition of nanoparticles increases the entropy generation rate, except at very low values of Re. This is attributed to the high viscosity of the base fluid e ethylene glycol. 4. The entropy generation for isothermal external fluid is between those for UHF and UWT thermal boundary conditions. The highest entropy is generated for UWT boundary condition while the lowest entropy is generated for UHF boundary condition. This places the UHF boundary condition as the most suitable boundary condition from the perspective of the second law, other parameters remaining the same. Further to the analysis carried out in this paper, research in future may be directed at minimizing the entropy generation rate with respect to the flow rate and the concentration of nanoparticles.
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