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Entropy generation between two vertical cylinders in the presence of MHD flow subjected to constant wall temperature
International Communications in Heat and Mass Transfer 44 (2013) 87–92
Contents lists available at SciVerse ScienceDirect
International Communications in Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ichmt
Entropy generation between two vertical cylinders in the presence of MHD flow subjected to constant wall temperature☆ Omid Mahian a,⁎, Hakan Oztop b, Ioan Pop c,⁎, Shohel Mahmud d, Somchai Wongwises e a
Young Researchers Club and Elites, Mashhad Branch, Islamic Azad University, Mashhad, Iran Department of Mechanical Engineering, Technology Faculty, Fırat University, Elazig, Turkey Department of Mathematics, Babeş-Bolyai University, R-400084 Cluj-Napoca, Romania d School of Engineering, University of Guelph, Guelph, Ontario, Canada e Fluid Mechanics, Thermal Engineering and Multiphase Flow Research Lab. (FUTURE), Department of Mechanical Engineering, Faculty of Engineering, King Mongkut's University of Technology Thonburi, Bangmod, Bangkok 10140, Thailand b c
a r t i c l e
i n f o
Available online 3 April 2013 Keywords: Entropy generation MHD flow Isothermal annulus Mixed convection
a b s t r a c t An analytical solution is presented on the entropy generation due to mixed convection between two isothermal cylinders where a transverse magnetic field is applied to the system. The governing equations in cylindrical coordinates are simplified and solved to obtain the distribution of entropy generation and the effects of MHD flow on it. The results for the entropy generation number (NS), the Bejan number (Be) and average entropy generation number (NS,ave) are presented for different values of the Hartmann numbers, radius ratios and a flow parameter, Gr/Re. The results show that the entropy generation decreases with increases in the magnetic field. In addition, it is found that with decreases in the radius ratio, the effects of MHD flow on the entropy generation are reduced. © 2013 Elsevier Ltd. All rights reserved.
1. Introduction There is a great deal of information available about the convective flow over bodies of various shapes and relevant literature can be found, for example, in the books by Bejan [1], and Martynenko and Khramtsov [2]. Entropy generation minimization is a well-known approach to optimize the performance of a thermal device. Entropy generation occurs in all of the thermal engineering systems. Bejan [3,4] investigated the different factors which are effective in the entropy generation in applied thermal engineering where available work (exergy) of a system is lost and destructed through the entropy production. Later on, many researchers focused on the entropy generation in different geometric arrangements, flow states, and thermal boundary conditions. Recently, Oztop and Al-Salem [5] presented a review of natural and mixed convection heat transfer for energy systems. The flow and heat transfer between two vertical cylinders in the presence of magnetic field has many important engineering applications. Some applications are found in industrial heat exchangers, microelectronic devices, cooling of nuclear reactors, petroleum equipment, and so forth. Therefore, before manufacturing such systems the analysis of the second law of thermodynamics (entropy generation) is necessary to have the best system design. For this purpose, an analysis of such systems using the simplified governing equations of a process, where reasonable
☆ Communicated by O.G. Martynenko. ⁎ Corresponding authors. E-mail addresses: [email protected] (I. Pop), [email protected] (O. Mahian). 0735-1933/$ – see front matter © 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.icheatmasstransfer.2013.03.005
assumptions are made, could be helpful to discover the important features of the complex process. Due to non-linearity nature of entropy generation problems, there are relatively few analytical works in this field. Here, some analytical works related to entropy generation between two cylinders in different situations (i.e., both cylinders are stationary, or one of them rotates) in which MHD flow has not considered, are mentioned in brief. Yilbas [6] by neglecting the irreversibility due to fluid friction and assuming a linear velocity profile presented an entropy generation analysis for a rotating outer cylinder and differentially heated isothermal boundary condition. Mahmud and Fraser [7,8] performed an entropy generation analysis within an annulus with isoflux and isothermal boundary conditions. They solved the simplified governing equations in cylindrical coordinates to obtain the dimensionless entropy generation and the Bejan number as a function of the parameters involved in the analysis. Tasnim and Mahmud [9,10] investigated the entropy production in a vertical annulus with a circular cross-section by considering the fully developed laminar flow and mixed convection heat transfer, analytically. Mirzazadeh et al. [11] studied the entropy generation between concentric rotating cylinders where the working fluid is a non-linear viscoelastic fluid. Mahian et al. [12] investigated analytically the entropy generation between two rotating cylinders using nanofluids with different volume fractions and isoflux boundary conditions. Mahian et al. [13] also studied the effects of uncertainties in the models presented for thermophysical properties of nanofluids on entropy generation. Making use of an external magnetic field is of considerable importance in many industrial applications, particularly as a control mechanism in material manufacturing [14]. In addition, the study of flow and heat transfer in
88
O. Mahian et al. / International Communications in Heat and Mass Transfer 44 (2013) 87–92
Nomenclature Be Br B0 E g k NF NH Nm NS P P* T T0 U U0 Q r R λ β μ SG
Bejan number Brinkman number constant magnetic flux density electric field gravitation acceleration thermal conductivity entropy generation number, fluid friction entropy generation number, heat transfer entropy generation number, MHD flow entropy generation number, total pressure dimensionless pressure temperature inlet temperature dimensionless velocity inlet velocity electric charge density radius dimensionless radius radius ratio (ri/r0) thermal expansion coefficient dynamic viscosity entropy generation rate
Greek symbols J electric current M Hartmann number Γn constants, n = 1,2,…. ρ density Π radius ratio (r0/ri) θ dimensionless temperature Ω dimensionless temperature difference ∀ volume
Subscripts I value at the inner cylinder O value at the outer cylinder
a closed cavity or a channel in the presence of the magnetic field is important because of engineering applications such as cooling of nuclear reactors, MHD marine propulsion, MHD micropumps, microelectronic devices, electronic packages, and so on. In the last decade, the study of MHD flow and its effect on entropy generation, in the related systems, received considerable attention because of its importance in industries, as mentioned above. Here, some works that investigated the effect of MHD flow on entropy generation for various flows and geometries have been reviewed. Salas et al. [15] analyzed the second law for MHD induction devices, such as electromagnetic pumps and electrical generators. Ibanez et al. [16] optimized the operation conditions of an alternate MHD generator based on the global entropy generation rate. Mahmud et al. [17] conducted an analysis to study the effects of transverse magnetic field on entropy generation inside a vertical channel made of two parallel plates by considering mixed convection inside the channel. Later on, Tasnim et al. [18] examined the same problem for porous media. Mahmud and Fraser [19] investigated analytically, the second law for mixed convection–radiation interaction in a vertical porous channel in the presence of MHD flow. In another work, Mahmud and Fraser [20] studied the problem of entropy generation in a fluid saturated porous cavity for laminar natural convection with MHD flow.
Al-Odat et al. [21] investigated numerically the effect of magnetic field on local entropy generation caused by steady two-dimensional laminar forced convection flow past a horizontal plate. Ibanez et al. [22] applied the entropy generation minimization method to optimize MHD flow between two infinite parallel walls with limited electrical conductivity. Mahmud and Fraser [23] presented a general relation for entropy generation for a single-plate thermoacoustic system that is exposed to a transverse magnetic field. Ibanez and Cuevas [24] considered a stationary buoyant MHD flow of a liquid metal immersed in a magnetic field through a vertical rectangular duct with thin conducting or insulating walls. Arikoglu et al. [25] investigated the effects of slip and Joule dissipation on the entropy generation in a single rotating disk in the presence of MHD flow. Aiboud and Saouli [26] applied the second law to analysis of the viscoelastic MHD flow over a stretching surface. Kolsi et al. [27] studied numerically the effect of MHD flow on the entropy generation of liquid metal laminar natural convection in a cubic cavity that is differentially heated. The effect of the magnetic field on entropy generation in a microchannel is investigated by Ibanez and Cuevas [28]. Recently, Mahian et al. [29] investigated the entropy generation between two isothermal rotating cylinders in the presence of MHD flow. They concluded that MHD flow increases the entropy generation. In the last work, Mahian et al. [30] studied the effects of nanofluids on entropy generation between two cylinders in the presence of a magnetic field. A review of the literature shows that there is no analytical work on entropy generation due to mixed convection in an annulus subjected to constant wall temperature and under magnetic field. The main aim of this work is to investigate the effects of magnetic field on the entropy generation in the annulus. The results for the entropy generation at different Hartmann numbers, radius ratios and the flow parameter Gr/Re are presented. 2. Problem formulation 2.1. Analysis according to the first law of thermodynamics Consider a steady laminar and fully developed flow where a Newtonian incompressible fluid enters a vertical annulus with an inlet velocity U0 and an inlet temperature T0. A constant magnetic field with strength of B0 is applied normally to the fluid flow direction. The schematic of the present problem is displayed in Fig. 1. Neglecting the velocity in r-direction, the heating due to magnetic field, and the use of Boussinesq approximation the momentum and energy equations become as follows (see Refs. [9, 10] for similar analysis):
−
" # 1 ∂P μ ∂2 V z 1 ∂V z σ B20 V z þ gβðT−T 0 Þ− ¼0 þ þ ρ ∂z ρ ∂r 2 r ∂r ρ
∂2 T 1 ∂T þ ¼0 ∂r 2 r ∂r
ð1Þ
ð2Þ
in the above relations ρ, μ, and σ are density, viscosity and fluid electrical conductivity of the working fluid, respectively. With the assumption of no slip between the walls of the cylinders, the velocity boundary conditions are given as follows: Vz ¼ 0 Vz ¼ 0
at at
r ¼ ri r ¼ r0 :
ð3Þ
The thermal boundary conditions are: T ¼ T i at r ¼ ri T ¼ T 0 at r ¼ r 0 :
ð4Þ
O. Mahian et al. / International Communications in Heat and Mass Transfer 44 (2013) 87–92
The following parameters are used for scaling the governing equations: r V U r r Pr z R ¼ ; U ¼ z ; Re ¼ 0 0 ; λ ¼ i ; P ¼ 0 ; Z ¼ ; M ¼ B0 r 0 r0 r0 U0 ν r0 U0μ
rffiffiffiffi σ ð5Þ μ
where M is the Hartmann number. In addition, the Grashof number and dimensionless temperature are as follows:
ð6Þ
Therefore, the dimensionless equations can be written as: ∂2 U 1 ∂U Gr ∂P 2 þ ¼− θþM Uþ 2 R ∂R Re ∂Z ∂R ∂2 θ 1 ∂θ þ ¼ 0: ∂R2 R ∂R
ð7Þ
ð8Þ
The boundary conditions in dimensional form become:velocity boundary conditions: U ¼ 0 at R ¼ λ U ¼ 0 at R ¼ 1 :
ð9Þ
Thermal boundary conditions: θ ¼ θi at R ¼ λ θ ¼ θ0 at R ¼ 1 :
2.2. Analysis according to the second law of thermodynamics The entropy generation rate in the presence of MHD flow can be written as follows: 000 k μ 1 2 S_ gen ¼ 2 ½∇T þ φ þ ½ðJ−QV Þ⋅ðE þ V BÞ T0 T0 T0
ð10Þ
ð11Þ
where: J ¼ σ ðE þ V BÞ:
3
T−T 0 gβΔTr0 θ¼ ; Gr ¼ ; ΔT ¼ T i −T 0 : ΔT kν2
89
ð12Þ
In the above relations, ϕ represents viscous dissipation, J is electric current, Q is electric charge density, V is velocity vector, E is the electric field, and B is the magnetic induction. Neglecting QV compared to J and disregarding E in comparison with V × B, the entropy generation rate, Eq. (11), is reduced in this case as: 000 k ∂T 2 μ ∂V z 2 σB20 2 þ þ V : S_ gen ¼ 2 T 0 ∂r T0 z T 0 ∂r
ð13Þ
The entropy generation number that is the dimensionless entropy generation rate becomes: 000
NS ¼
S_ gen μU 20 T 0 r 20
¼
2 2 Ω ∂θ ∂U 2 2 þ þ M U ¼ NH þ NF þ NM Br ∂R ∂R
ð14Þ
where NH, NF, and NM on the right hand of the equation are the irreversibilities due to heat transfer, fluid friction, and magnetic field. Also, the Brinkman number (Br) and the parameter Ω are defined as follows: 2
Br ¼
U0μ ΔT ;Ω ¼ : kΔT T0
ð15Þ
To determine the irreversibility distribution, Bejan number (Be) that is the ratio of entropy generation due to heat transfer to the overall entropy generation [31], is defined as: Be ¼
NH : NH þ NF þ NM
ð16Þ
The average (overall) entropy generation number is given as: NS;ave ¼
1 ∫ N d∀: ∀∀ S
ð17Þ
3. Solution of the problem 3.1. Velocity and temperature distributions Eqs. (7) and (8) are coupled via the buoyancy term in the momentum equation. Therefore, firstly the energy equation should be solved. The solution of the energy equation after applying the boundary conditions is as: θðRÞ ¼
θi −θo lnðRÞ þ θo : lnðλÞ
ð18Þ
Now, the velocity distribution is obtained easily as:
Fig. 1. Schematic of the present problem.
U ðRÞ ¼ Γ 1 I 0 ðMRÞ þ Γ 2 K 0 ðMRÞ þ Γ 3 lnðRÞ þ Γ 4
ð19Þ
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O. Mahian et al. / International Communications in Heat and Mass Transfer 44 (2013) 87–92
where I1(MR) is the modified Bessel function of the first kind, of order 1, and K1(MR) as the modified Bessel function of the second kind, of order 1. The constants Γ1 − Γ4 are as follows:
Γ2 Γ3 Γ4
25
15
ð20Þ 10
5 0.5
3.2. Local entropy generation The local entropy generation can be obtained based on Eq. (14) and the velocity and temperature distributions. Therefore, the entropy generation number can be determined by the following relation: NS ¼
M=1 M=3 M=5 M = 10
20
Ns
Γ1
ðGr=ReÞ½θo K 0 ðMλÞ−θi K 0 ðMÞ þ ð∂P =∂Z Þ½K 0 ðM Þ−K 0 ðMλÞ ¼ M 2 ½K 0 ðM ÞI0 ðMλÞ−K 0 ðMλÞI 0 ðMÞ ðGr=ReÞ½θo I 0 ðMλÞ−θi I0 ðM Þ þ ð∂P =∂Z Þ½I0 ðMÞ−I0 ðMλÞ ¼− M2 ½K 0 ðMÞI0 ðMλÞ−K 0 ðMλÞI0 ðM Þ ðGr=ReÞðθi −θo Þ ¼ M2 lnðλÞ ½ðGr=ReÞθo −∂P =∂Z ¼ M2
A
B
Ω θi −θo 2 Γ 2 þ Γ 1 MI1 ðMRÞ−Γ 2 MK 1 ðMRÞ þ 3 þ Br lnðλÞR R
3.3. Average entropy generation number
1
Be 0.8
M=1 M=3 M=5 M = 10
The average entropy generation number is calculated based on Eq. (17): 0.5
2 ∫ NS RdR: 1−λ2 λ
0.6
0.7
0.8
0.9
1
R
1
NS;ave ¼
1
R
0.9
M2 ðΓ 1 I0 ðMRÞ þ Γ 2 K 0 ðMRÞ þ Γ 3 lnðRÞ þ Γ 4 Þ2 ¼ NH þ NF þ NM : ð21Þ
0.75
ð22Þ
4. Results and discussions Before presenting the results, it should be noted that in this work only the effects of flow parameters including, Gr/Re, M and the geometric parameter of radius ratio on entropy generation are investigated. Fig. 2 shows the effects of MHD flow on the entropy generation number and Bejan number for θi = 1, θ0 = 0.5, Ω/Br = 10, dP*/ dZ = − 0.1 and Gr/Re = 10. As seen, an increase in the Hartmann number leads to a decrease in the entropy generation number near the walls of the cylinders whereas the value of NS increases in the middle of the two cylinders. In this case, although the contribution of the magnetic field to entropy generation is increased a little, the magnetic field reduces the contribution of fluid friction to entropy generation with the lowering of the shear stresses and hence the velocity gradients near the walls. In the middle of the annulus where the effects of walls on the flow diminish, the entropy generation increases with an increase in the Hartmann number. The distribution of the Bejan number Be shows that the contribution of viscous dissipations (NF) near the walls decreases with an increase in the Hartmann number. It should be noted that based on Eq. (14), the magnetic field has no effect on the temperature field and hence NH remains constant. Therefore, according to the definition of the Bejan number (see Eq. (16)), with the reduction of NF, the Bejan number will increase. Fig. 3 shows the variation of the entropy generation and the Bejan number in the annulus at different values of Gr/Re. The parameter of Gr/Re represents the ratio of free convection to forced convection. When Gr/Re = 0.1, the forced convection is dominant on free convection. Therefore, because of the enhancement of heat transfer in forced convection rather than free convection, the
Fig. 2. Effects of MHD flow on (a) entropy generation number and (b) Bejan number.
irreversibilities are decreased in this case. The distribution of the Bejan number shows that the irreversibility due to heat transfer is the highest in the middle of annulus for any value of Gr/Re. Fig. 4 shows the effects of MHD flow on the average entropy generation number at different values of Gr/Re. It is observed that with an increase in the magnetic field the average entropy generation number remains nearly constant for Gr/Re = 0.1. However, in high values of Gr/Re, the entropy generation decreases with increases in the Hartmann number. The rate of this decrease is greater for higher amounts of Gr/Re. An important issue in the design of such type of heat exchanger is the selection of optimum radius ratios. The effects of MHD flow on average entropy generation for different values of radius ratios (Π = 1/λ = r0/ri) are presented in Fig. 5 for 1.5 ≤ Π ≤ 6 and Gr/Re = 10. It is observed that by increasing the radius ratio the entropy generation is decreased due to decreases of gradients in the gap. As shown, for smaller radius ratios the MHD flow effects on the average entropy generation are not visible. In other words, the decrease in the entropy generation is higher in magnitude for greater radius ratios. 5. Conclusion An entropy generation analysis is conducted to find the effects of MHD flow on the entropy generation due to a mixed convection between two vertical cylinders. The equations of momentum and energy in cylindrical coordinates are simplified and solved analytically. The results show that an increase in the magnetic field results in a decrease in the entropy generation. It is also found that the decrease in the entropy generation is higher for greater radius ratios.
O. Mahian et al. / International Communications in Heat and Mass Transfer 44 (2013) 87–92
A
91
50
M = 0.1 M=3 M=5 M = 10
20
Gr / Re = 0.1 Gr / Re = 10 Gr / Re = 30
40
Ns,ave
15
Ns
30
10 20 5 10 2
3
4
5
6
Π
0 0.5
0.75
1 Fig. 5. Effects MHD flow on the average entropy generation at different radius ratios (Π = 1/λ).
R
B 1
Gr / Re = 0.1
References
Be
Gr / Re = 10
0.75
Gr / Re = 30
0.5 0.5
0.75
1
R Fig. 3. Effects of Gr/Re on (a) entropy generation number and (b) Bejan number.
Acknowledgment The first and fifth authors would like to thank the Thailand Research Fund and the National Research University Project for supporting this study.
14
Ns,ave
Gr / Re = 20
12
Gr / Re = 15
Gr / Re = 10 10
Gr / Re = 0.1 2
4
6
8
10
M Fig. 4. Effects of MHD flow on average entropy generation number at different Gr/Re.
[1] A. Bejan, Convection Heat Transfer, 2nd edition Wiley, New York, 1995. [2] O.G. Martynenko, P.P. Khramtsov, Free-convective Heat Transfer, Springer, New York, 2005. [3] A. Bejan, Second-law analysis in heat transfer and thermal design, Advances in Heat Transfer 15 (1982) 1–58. [4] A. Bejan, Entropy Generation Minimization, CRC Press, Boca Raton, NY, 1996. [5] H.F. Oztop, K. Al-Salem, A review on entropy generation in natural and mixed convection heat transfer for energy systems, Renewable and Sustainable Energy Reviews 16 (2012) 911–920. [6] B.S. Yilbas, Entropy analysis of concentric annuli with rotating outer cylinder, International Journal of Exergy 1 (2001) 60–66. [7] S. Mahmud, R.A. Fraser, Second law analysis of heat transfer and fluid flow inside a cylindrical annular space, International Journal of Exergy 2 (2002) 322–329. [8] S. Mahmud, R.A. Fraser, Analysis of entropy generation inside concentric cylindrical annuli with relative rotation, International Journal of Thermal Sciences 42 (2003) 513–521. [9] S.H. Tasnim, S. Mahmud, Mixed convection and entropy generation in a vertical annular space, International Journal of Exergy 2 (2002) 373–379. [10] S.H. Tasnim, S. Mahmud, Entropy generation in a vertical concentric channel with temperature dependent viscosity, International Communications in Heat and Mass Transfer 29 (2002) 907–918. [11] M. Mirzazadeh, A. Shafaei, F. Rashidi, Entropy analysis for non-linear viscoelastic fluid in concentric rotating cylinders, International Journal of Thermal Sciences 47 (2008) 1701–1711. [12] O. Mahian, S. Mahmud, S. Zeinali Heris, Analysis of entropy generation between co-rotating cylinders using nanofluids, Energy 44 (2012) 438–446. [13] O. Mahian, S. Mahmud, S. Zeinali Heris, Effect of uncertainties in physical properties on entropy generation between two rotating cylinders with nanofluids, ASME Journal of Heat Transfer 134 (10) (2012), [art. no. 101704]. [14] A.E. Jery, N. Hidouri, M. Magherbi, A.B. Brahim, Effect of an external oriented magnetic field on entropy generation in natural convection, Entropy 12 (2010) 1391–1417. [15] H. Salas, S. Cuevas, M.L. Haro, Entropy generation analysis of magnetohydrodynamic induction devices, Journal of Physics D: Applied Physics 32 (1999) 2605. [16] G. Ibanez, S. Cuevas, M.L. Haro, Optimization analysis of an alternate magnetohydrodynamic generator, Energy Conversion and Management 43 (2002) 1757–1771. [17] S. Mahmud, S.H. Tasnim, M.A.H. Mamun, Thermodynamic analysis of mixed convection in a channel with transverse hydromagnetic effect, International Journal of Thermal Sciences 42 (2003) 731–740. [18] S.H. Tasnim, M. Shohel, M.A.H. Mamun, Entropy generation in a porous channel with hydromagnetic effect, Energy 2 (2002) 300–308. [19] S. Mahmud, R.A. Fraser, Mixed convection–radiation interaction in a vertical porous channel: entropy generation, Energy 28 (2003) 1557–1577. [20] S. Mahmud, R.A. Fraser, Magnetohydrodynamic free convection and entropy generation in a square porous cavity, International Journal of Heat and Mass Transfer 47 (2004) 3245–3256. [21] M. Odat, R.A. Damseh, M. Nimr, Effect of magnetic field on entropy generation due to laminar forced convection past a horizontal flat plate, Entropy 4 (2004) 293–303. [22] G. Ibanez, S. Cuevas, M.L. Haro, Optimization of a magnetohydrodynamic flow based on the entropy generation minimization method, International Communications in Heat and Mass Transfer 33 (2006) 295–301. [23] S. Mahmud, R.A. Fraser, The thermagoustic irreversibility for a single-plate thermoacoustic system, International Journal of Heat and Mass Transfer 49 (2006) 3448–3461. [24] G. Ibanez, S. Cuevas, Optimum wall conductance ratio in magnetoconvective flow in a long vertical rectangular duct, International Journal of Thermal Sciences 47 (2008) 1012–1019.
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[25] A. Arikoglu, I. Ozkol, G. Komurgoz, Effect of slip on entropy generation in a single rotating disk in MHD flow, Applied Energy 85 (2008) 1225–1236. [26] S. AIboud, S. Saouli, Entropy analysis for viscoelastic magnetohydrodynamic flow over a stretching surface, International Journal of Non-Linear Mechanics 45 (2010) 482–489. [27] L. Kolsi, A. Abidi, M.N. Borjini, H.B. Aissia, The effect of an external magnetic field on the entropy generation in three dimensional natural convection, Thermal Science 14 (2010) 341–352. [28] G. Ibanez, S. Cuevas, Entropy generation minimization of a MHD (magnetohydrodynamic) flow in a microchannel, Energy 35 (2010) 4149–4155.
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Contents lists available at SciVerse ScienceDirect
International Communications in Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ichmt
Entropy generation between two vertical cylinders in the presence of MHD flow subjected to constant wall temperature☆ Omid Mahian a,⁎, Hakan Oztop b, Ioan Pop c,⁎, Shohel Mahmud d, Somchai Wongwises e a
Young Researchers Club and Elites, Mashhad Branch, Islamic Azad University, Mashhad, Iran Department of Mechanical Engineering, Technology Faculty, Fırat University, Elazig, Turkey Department of Mathematics, Babeş-Bolyai University, R-400084 Cluj-Napoca, Romania d School of Engineering, University of Guelph, Guelph, Ontario, Canada e Fluid Mechanics, Thermal Engineering and Multiphase Flow Research Lab. (FUTURE), Department of Mechanical Engineering, Faculty of Engineering, King Mongkut's University of Technology Thonburi, Bangmod, Bangkok 10140, Thailand b c
a r t i c l e
i n f o
Available online 3 April 2013 Keywords: Entropy generation MHD flow Isothermal annulus Mixed convection
a b s t r a c t An analytical solution is presented on the entropy generation due to mixed convection between two isothermal cylinders where a transverse magnetic field is applied to the system. The governing equations in cylindrical coordinates are simplified and solved to obtain the distribution of entropy generation and the effects of MHD flow on it. The results for the entropy generation number (NS), the Bejan number (Be) and average entropy generation number (NS,ave) are presented for different values of the Hartmann numbers, radius ratios and a flow parameter, Gr/Re. The results show that the entropy generation decreases with increases in the magnetic field. In addition, it is found that with decreases in the radius ratio, the effects of MHD flow on the entropy generation are reduced. © 2013 Elsevier Ltd. All rights reserved.
1. Introduction There is a great deal of information available about the convective flow over bodies of various shapes and relevant literature can be found, for example, in the books by Bejan [1], and Martynenko and Khramtsov [2]. Entropy generation minimization is a well-known approach to optimize the performance of a thermal device. Entropy generation occurs in all of the thermal engineering systems. Bejan [3,4] investigated the different factors which are effective in the entropy generation in applied thermal engineering where available work (exergy) of a system is lost and destructed through the entropy production. Later on, many researchers focused on the entropy generation in different geometric arrangements, flow states, and thermal boundary conditions. Recently, Oztop and Al-Salem [5] presented a review of natural and mixed convection heat transfer for energy systems. The flow and heat transfer between two vertical cylinders in the presence of magnetic field has many important engineering applications. Some applications are found in industrial heat exchangers, microelectronic devices, cooling of nuclear reactors, petroleum equipment, and so forth. Therefore, before manufacturing such systems the analysis of the second law of thermodynamics (entropy generation) is necessary to have the best system design. For this purpose, an analysis of such systems using the simplified governing equations of a process, where reasonable
☆ Communicated by O.G. Martynenko. ⁎ Corresponding authors. E-mail addresses: [email protected] (I. Pop), [email protected] (O. Mahian). 0735-1933/$ – see front matter © 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.icheatmasstransfer.2013.03.005
assumptions are made, could be helpful to discover the important features of the complex process. Due to non-linearity nature of entropy generation problems, there are relatively few analytical works in this field. Here, some analytical works related to entropy generation between two cylinders in different situations (i.e., both cylinders are stationary, or one of them rotates) in which MHD flow has not considered, are mentioned in brief. Yilbas [6] by neglecting the irreversibility due to fluid friction and assuming a linear velocity profile presented an entropy generation analysis for a rotating outer cylinder and differentially heated isothermal boundary condition. Mahmud and Fraser [7,8] performed an entropy generation analysis within an annulus with isoflux and isothermal boundary conditions. They solved the simplified governing equations in cylindrical coordinates to obtain the dimensionless entropy generation and the Bejan number as a function of the parameters involved in the analysis. Tasnim and Mahmud [9,10] investigated the entropy production in a vertical annulus with a circular cross-section by considering the fully developed laminar flow and mixed convection heat transfer, analytically. Mirzazadeh et al. [11] studied the entropy generation between concentric rotating cylinders where the working fluid is a non-linear viscoelastic fluid. Mahian et al. [12] investigated analytically the entropy generation between two rotating cylinders using nanofluids with different volume fractions and isoflux boundary conditions. Mahian et al. [13] also studied the effects of uncertainties in the models presented for thermophysical properties of nanofluids on entropy generation. Making use of an external magnetic field is of considerable importance in many industrial applications, particularly as a control mechanism in material manufacturing [14]. In addition, the study of flow and heat transfer in
88
O. Mahian et al. / International Communications in Heat and Mass Transfer 44 (2013) 87–92
Nomenclature Be Br B0 E g k NF NH Nm NS P P* T T0 U U0 Q r R λ β μ SG
Bejan number Brinkman number constant magnetic flux density electric field gravitation acceleration thermal conductivity entropy generation number, fluid friction entropy generation number, heat transfer entropy generation number, MHD flow entropy generation number, total pressure dimensionless pressure temperature inlet temperature dimensionless velocity inlet velocity electric charge density radius dimensionless radius radius ratio (ri/r0) thermal expansion coefficient dynamic viscosity entropy generation rate
Greek symbols J electric current M Hartmann number Γn constants, n = 1,2,…. ρ density Π radius ratio (r0/ri) θ dimensionless temperature Ω dimensionless temperature difference ∀ volume
Subscripts I value at the inner cylinder O value at the outer cylinder
a closed cavity or a channel in the presence of the magnetic field is important because of engineering applications such as cooling of nuclear reactors, MHD marine propulsion, MHD micropumps, microelectronic devices, electronic packages, and so on. In the last decade, the study of MHD flow and its effect on entropy generation, in the related systems, received considerable attention because of its importance in industries, as mentioned above. Here, some works that investigated the effect of MHD flow on entropy generation for various flows and geometries have been reviewed. Salas et al. [15] analyzed the second law for MHD induction devices, such as electromagnetic pumps and electrical generators. Ibanez et al. [16] optimized the operation conditions of an alternate MHD generator based on the global entropy generation rate. Mahmud et al. [17] conducted an analysis to study the effects of transverse magnetic field on entropy generation inside a vertical channel made of two parallel plates by considering mixed convection inside the channel. Later on, Tasnim et al. [18] examined the same problem for porous media. Mahmud and Fraser [19] investigated analytically, the second law for mixed convection–radiation interaction in a vertical porous channel in the presence of MHD flow. In another work, Mahmud and Fraser [20] studied the problem of entropy generation in a fluid saturated porous cavity for laminar natural convection with MHD flow.
Al-Odat et al. [21] investigated numerically the effect of magnetic field on local entropy generation caused by steady two-dimensional laminar forced convection flow past a horizontal plate. Ibanez et al. [22] applied the entropy generation minimization method to optimize MHD flow between two infinite parallel walls with limited electrical conductivity. Mahmud and Fraser [23] presented a general relation for entropy generation for a single-plate thermoacoustic system that is exposed to a transverse magnetic field. Ibanez and Cuevas [24] considered a stationary buoyant MHD flow of a liquid metal immersed in a magnetic field through a vertical rectangular duct with thin conducting or insulating walls. Arikoglu et al. [25] investigated the effects of slip and Joule dissipation on the entropy generation in a single rotating disk in the presence of MHD flow. Aiboud and Saouli [26] applied the second law to analysis of the viscoelastic MHD flow over a stretching surface. Kolsi et al. [27] studied numerically the effect of MHD flow on the entropy generation of liquid metal laminar natural convection in a cubic cavity that is differentially heated. The effect of the magnetic field on entropy generation in a microchannel is investigated by Ibanez and Cuevas [28]. Recently, Mahian et al. [29] investigated the entropy generation between two isothermal rotating cylinders in the presence of MHD flow. They concluded that MHD flow increases the entropy generation. In the last work, Mahian et al. [30] studied the effects of nanofluids on entropy generation between two cylinders in the presence of a magnetic field. A review of the literature shows that there is no analytical work on entropy generation due to mixed convection in an annulus subjected to constant wall temperature and under magnetic field. The main aim of this work is to investigate the effects of magnetic field on the entropy generation in the annulus. The results for the entropy generation at different Hartmann numbers, radius ratios and the flow parameter Gr/Re are presented. 2. Problem formulation 2.1. Analysis according to the first law of thermodynamics Consider a steady laminar and fully developed flow where a Newtonian incompressible fluid enters a vertical annulus with an inlet velocity U0 and an inlet temperature T0. A constant magnetic field with strength of B0 is applied normally to the fluid flow direction. The schematic of the present problem is displayed in Fig. 1. Neglecting the velocity in r-direction, the heating due to magnetic field, and the use of Boussinesq approximation the momentum and energy equations become as follows (see Refs. [9, 10] for similar analysis):
−
" # 1 ∂P μ ∂2 V z 1 ∂V z σ B20 V z þ gβðT−T 0 Þ− ¼0 þ þ ρ ∂z ρ ∂r 2 r ∂r ρ
∂2 T 1 ∂T þ ¼0 ∂r 2 r ∂r
ð1Þ
ð2Þ
in the above relations ρ, μ, and σ are density, viscosity and fluid electrical conductivity of the working fluid, respectively. With the assumption of no slip between the walls of the cylinders, the velocity boundary conditions are given as follows: Vz ¼ 0 Vz ¼ 0
at at
r ¼ ri r ¼ r0 :
ð3Þ
The thermal boundary conditions are: T ¼ T i at r ¼ ri T ¼ T 0 at r ¼ r 0 :
ð4Þ
O. Mahian et al. / International Communications in Heat and Mass Transfer 44 (2013) 87–92
The following parameters are used for scaling the governing equations: r V U r r Pr z R ¼ ; U ¼ z ; Re ¼ 0 0 ; λ ¼ i ; P ¼ 0 ; Z ¼ ; M ¼ B0 r 0 r0 r0 U0 ν r0 U0μ
rffiffiffiffi σ ð5Þ μ
where M is the Hartmann number. In addition, the Grashof number and dimensionless temperature are as follows:
ð6Þ
Therefore, the dimensionless equations can be written as: ∂2 U 1 ∂U Gr ∂P 2 þ ¼− θþM Uþ 2 R ∂R Re ∂Z ∂R ∂2 θ 1 ∂θ þ ¼ 0: ∂R2 R ∂R
ð7Þ
ð8Þ
The boundary conditions in dimensional form become:velocity boundary conditions: U ¼ 0 at R ¼ λ U ¼ 0 at R ¼ 1 :
ð9Þ
Thermal boundary conditions: θ ¼ θi at R ¼ λ θ ¼ θ0 at R ¼ 1 :
2.2. Analysis according to the second law of thermodynamics The entropy generation rate in the presence of MHD flow can be written as follows: 000 k μ 1 2 S_ gen ¼ 2 ½∇T þ φ þ ½ðJ−QV Þ⋅ðE þ V BÞ T0 T0 T0
ð10Þ
ð11Þ
where: J ¼ σ ðE þ V BÞ:
3
T−T 0 gβΔTr0 θ¼ ; Gr ¼ ; ΔT ¼ T i −T 0 : ΔT kν2
89
ð12Þ
In the above relations, ϕ represents viscous dissipation, J is electric current, Q is electric charge density, V is velocity vector, E is the electric field, and B is the magnetic induction. Neglecting QV compared to J and disregarding E in comparison with V × B, the entropy generation rate, Eq. (11), is reduced in this case as: 000 k ∂T 2 μ ∂V z 2 σB20 2 þ þ V : S_ gen ¼ 2 T 0 ∂r T0 z T 0 ∂r
ð13Þ
The entropy generation number that is the dimensionless entropy generation rate becomes: 000
NS ¼
S_ gen μU 20 T 0 r 20
¼
2 2 Ω ∂θ ∂U 2 2 þ þ M U ¼ NH þ NF þ NM Br ∂R ∂R
ð14Þ
where NH, NF, and NM on the right hand of the equation are the irreversibilities due to heat transfer, fluid friction, and magnetic field. Also, the Brinkman number (Br) and the parameter Ω are defined as follows: 2
Br ¼
U0μ ΔT ;Ω ¼ : kΔT T0
ð15Þ
To determine the irreversibility distribution, Bejan number (Be) that is the ratio of entropy generation due to heat transfer to the overall entropy generation [31], is defined as: Be ¼
NH : NH þ NF þ NM
ð16Þ
The average (overall) entropy generation number is given as: NS;ave ¼
1 ∫ N d∀: ∀∀ S
ð17Þ
3. Solution of the problem 3.1. Velocity and temperature distributions Eqs. (7) and (8) are coupled via the buoyancy term in the momentum equation. Therefore, firstly the energy equation should be solved. The solution of the energy equation after applying the boundary conditions is as: θðRÞ ¼
θi −θo lnðRÞ þ θo : lnðλÞ
ð18Þ
Now, the velocity distribution is obtained easily as:
Fig. 1. Schematic of the present problem.
U ðRÞ ¼ Γ 1 I 0 ðMRÞ þ Γ 2 K 0 ðMRÞ þ Γ 3 lnðRÞ þ Γ 4
ð19Þ
90
O. Mahian et al. / International Communications in Heat and Mass Transfer 44 (2013) 87–92
where I1(MR) is the modified Bessel function of the first kind, of order 1, and K1(MR) as the modified Bessel function of the second kind, of order 1. The constants Γ1 − Γ4 are as follows:
Γ2 Γ3 Γ4
25
15
ð20Þ 10
5 0.5
3.2. Local entropy generation The local entropy generation can be obtained based on Eq. (14) and the velocity and temperature distributions. Therefore, the entropy generation number can be determined by the following relation: NS ¼
M=1 M=3 M=5 M = 10
20
Ns
Γ1
ðGr=ReÞ½θo K 0 ðMλÞ−θi K 0 ðMÞ þ ð∂P =∂Z Þ½K 0 ðM Þ−K 0 ðMλÞ ¼ M 2 ½K 0 ðM ÞI0 ðMλÞ−K 0 ðMλÞI 0 ðMÞ ðGr=ReÞ½θo I 0 ðMλÞ−θi I0 ðM Þ þ ð∂P =∂Z Þ½I0 ðMÞ−I0 ðMλÞ ¼− M2 ½K 0 ðMÞI0 ðMλÞ−K 0 ðMλÞI0 ðM Þ ðGr=ReÞðθi −θo Þ ¼ M2 lnðλÞ ½ðGr=ReÞθo −∂P =∂Z ¼ M2
A
B
Ω θi −θo 2 Γ 2 þ Γ 1 MI1 ðMRÞ−Γ 2 MK 1 ðMRÞ þ 3 þ Br lnðλÞR R
3.3. Average entropy generation number
1
Be 0.8
M=1 M=3 M=5 M = 10
The average entropy generation number is calculated based on Eq. (17): 0.5
2 ∫ NS RdR: 1−λ2 λ
0.6
0.7
0.8
0.9
1
R
1
NS;ave ¼
1
R
0.9
M2 ðΓ 1 I0 ðMRÞ þ Γ 2 K 0 ðMRÞ þ Γ 3 lnðRÞ þ Γ 4 Þ2 ¼ NH þ NF þ NM : ð21Þ
0.75
ð22Þ
4. Results and discussions Before presenting the results, it should be noted that in this work only the effects of flow parameters including, Gr/Re, M and the geometric parameter of radius ratio on entropy generation are investigated. Fig. 2 shows the effects of MHD flow on the entropy generation number and Bejan number for θi = 1, θ0 = 0.5, Ω/Br = 10, dP*/ dZ = − 0.1 and Gr/Re = 10. As seen, an increase in the Hartmann number leads to a decrease in the entropy generation number near the walls of the cylinders whereas the value of NS increases in the middle of the two cylinders. In this case, although the contribution of the magnetic field to entropy generation is increased a little, the magnetic field reduces the contribution of fluid friction to entropy generation with the lowering of the shear stresses and hence the velocity gradients near the walls. In the middle of the annulus where the effects of walls on the flow diminish, the entropy generation increases with an increase in the Hartmann number. The distribution of the Bejan number Be shows that the contribution of viscous dissipations (NF) near the walls decreases with an increase in the Hartmann number. It should be noted that based on Eq. (14), the magnetic field has no effect on the temperature field and hence NH remains constant. Therefore, according to the definition of the Bejan number (see Eq. (16)), with the reduction of NF, the Bejan number will increase. Fig. 3 shows the variation of the entropy generation and the Bejan number in the annulus at different values of Gr/Re. The parameter of Gr/Re represents the ratio of free convection to forced convection. When Gr/Re = 0.1, the forced convection is dominant on free convection. Therefore, because of the enhancement of heat transfer in forced convection rather than free convection, the
Fig. 2. Effects of MHD flow on (a) entropy generation number and (b) Bejan number.
irreversibilities are decreased in this case. The distribution of the Bejan number shows that the irreversibility due to heat transfer is the highest in the middle of annulus for any value of Gr/Re. Fig. 4 shows the effects of MHD flow on the average entropy generation number at different values of Gr/Re. It is observed that with an increase in the magnetic field the average entropy generation number remains nearly constant for Gr/Re = 0.1. However, in high values of Gr/Re, the entropy generation decreases with increases in the Hartmann number. The rate of this decrease is greater for higher amounts of Gr/Re. An important issue in the design of such type of heat exchanger is the selection of optimum radius ratios. The effects of MHD flow on average entropy generation for different values of radius ratios (Π = 1/λ = r0/ri) are presented in Fig. 5 for 1.5 ≤ Π ≤ 6 and Gr/Re = 10. It is observed that by increasing the radius ratio the entropy generation is decreased due to decreases of gradients in the gap. As shown, for smaller radius ratios the MHD flow effects on the average entropy generation are not visible. In other words, the decrease in the entropy generation is higher in magnitude for greater radius ratios. 5. Conclusion An entropy generation analysis is conducted to find the effects of MHD flow on the entropy generation due to a mixed convection between two vertical cylinders. The equations of momentum and energy in cylindrical coordinates are simplified and solved analytically. The results show that an increase in the magnetic field results in a decrease in the entropy generation. It is also found that the decrease in the entropy generation is higher for greater radius ratios.
O. Mahian et al. / International Communications in Heat and Mass Transfer 44 (2013) 87–92
A
91
50
M = 0.1 M=3 M=5 M = 10
20
Gr / Re = 0.1 Gr / Re = 10 Gr / Re = 30
40
Ns,ave
15
Ns
30
10 20 5 10 2
3
4
5
6
Π
0 0.5
0.75
1 Fig. 5. Effects MHD flow on the average entropy generation at different radius ratios (Π = 1/λ).
R
B 1
Gr / Re = 0.1
References
Be
Gr / Re = 10
0.75
Gr / Re = 30
0.5 0.5
0.75
1
R Fig. 3. Effects of Gr/Re on (a) entropy generation number and (b) Bejan number.
Acknowledgment The first and fifth authors would like to thank the Thailand Research Fund and the National Research University Project for supporting this study.
14
Ns,ave
Gr / Re = 20
12
Gr / Re = 15
Gr / Re = 10 10
Gr / Re = 0.1 2
4
6
8
10
M Fig. 4. Effects of MHD flow on average entropy generation number at different Gr/Re.
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