Investigation of Nanofluid MHD Flow and Heat Transfer in a Channel

Size: px

Start display at page:

Download "Investigation of Nanofluid MHD Flow and Heat Transfer in a Channel"

Clemence Owens
6 years ago
Views:

1 Copyright 2014 by American Scientific Publishers All rights reserved. Printed in the United States of America Journal of Advanced Physics Vol. 3, pp. 1 11, Investigation of Nanofluid MHD Flow and Heat Transfer in a Channel M. Fakou 1,,D.D.Ganji 2, and M. Abbasi 1 1 Department of Mechanical Engineering, Sari Branch, Islamic Azad University, Sari, Iran 2 Department of Mechanical Engineering, Babol University of Technology, Babol, Iran In this study, laminar fluid flow and heat transfer in channel with permeable walls in the presence of a transverse magnetic field is investigated. Homotopy perturbation method HPM has been employed for computing approximate solutions of nonlinear differential equations governing the problem. We have tried to show reliability and performance of the present method compared with the numerical method Runge-Kutta fourth-rate to solve this problem. The influence of the four dimensionless numbers: the Hartmann number, Reynolds number, Prandtl number and Eckert number on the non-dimensional velocity and temperature profiles have been investigated. The results show Homotopy perturbation method is very close to numerically method. In general, increasing the Reynolds number is increases the nanofluid flow velocity in the channel and the maximum amount of temperature increase and by applied magnetic field Ha, velocity and the maximum amount of temperature in the channel is reduces. Increasing Prandtl number and Eckert, respectively, the maximum value of theta increases and decreases. KEYWORDS: Homotopy Perturbation Method HPM, Laminar Viscous Flow, Heat Transfer, Uniform Magnetic Field. 1. INTRODUCTION Flow problem in a porous tube or channel received much attention in recent years due to its various applications in medical engineering, for example, in the dialysis of blood in artificial kidney, 1 in the flow of blood in the capillaries, 2 in the flow in blood oxygenators, 3 as well as in many other engineering areas such as the design of filters, 4 in transpiration cooling boundary layer control 5 and gaseous diffusion. 6 In 1953, Berman 7 described an exact solution of the Navier-Stokes equation for steady two-dimensional laminar flow of a viscous, incompressible fluid in a channel with parallel, rigid, porous walls driven by uniform, steady suction or injection at the walls. This mass transfer is paramount in some industrial processes. More recently, Chandran and Sacheti 8 analyzed the effects of a magnetic field on the thermodynamic flow past a continuously moving porous plate. Slow viscous flow problem in a semi-porous channel in the presence of transverse magnetic field is investigated by Sheikholeslami et al. 9 They show that the method is asymptotically optimal Homotopy a powerful method for solving nonlinear differential equations, such as the problem. Soleimani et al. 10 studied of natural convection heat Author to whom correspondence should be addressed. s: mehdi_fakour@yahoo.com, mehdi_fakoor8@yahoo.com Received: 13 October 2014 Accepted: 23 October 2014 transfer in an enclosure filled mid-loop with nanofluid using the Control Volume based Finite Element Method. They founded turn angle has a significant effect on flow lines, isotherms and local Nusselt number is maximum or minimum values. Sheikholeslami et al. 11 investigated the flow of nanofluid and heat transfer characteristics between two horizontal plates in a rotating system. Their results showed that for suction and injection, the heat transfer rate increases with the nanoparticle volume fraction, Reynolds number, and parameter injection/suction increases and then decreases with the strength of the spin parameter. Steady magneto hydrodynamic free convection boundary layer flow past a vertical semi-infinite flat plate embedded in water filled with a nanofluid has been theoretically studied by Hamad et al. 12 They found that copper and silver nanoparticles have proved that the highest cooling performance for this problem. Natural convection of a non- Newtonian copper-water nanofluid between two infinite parallel vertical flat plates was investigated by Domairry et al. 13 They concluded that as the size of nanoparticles increases, the boundary layer thickness increases, the thermal boundary layer thickness decreases. Sheikholeslami et al. 14 studied the natural convection in a concentric annulus between a cold outer square and heated inner circular cylinders in the presence of static radial magnetic field. Sheikholeslami et al. 15 performed a numerical analysis for natural convection heat transfer of Cu-water nanofluid in a cold outer circular enclosure containing J. Adv. Phys. 2014, Vol. 3, No /2014/3/001/011 doi: /jap

2 Investigation of Nanofluid MHD Flow and Heat Transfer in a Channel Fakou et al. a hot inner sinusoidal circular cylinder in the presence of horizontal magnetic field using the Control Volume based Finite Element Method. They concluded that in the absence of a magnetic field, increasing the Rayleigh number increases, while the opposite trend was observed in the presence of a magnetic field decreases. Sheikholeslami et al. 16 studied the effects of magnetic field and nanoparticle on the Jeffery Hamel flow using Adomian decomposition method. They showed that increasing Hartmann number will lead to back flow reduction. Recently several authors investigated about nanofluid flow and heat transfer. The main goal of this paper is to examine the laminar nanofluid flow in channel with permeable walls in the presence of transverse magnetic field using Homotopy perturbation method. Effective volume fraction nanofluid, Hartmann number, Reynolds number, Prandtl number, and Eckert number on the velocity and temperature considered. In addition to speed and temperature for different structures nanofluid copper and silver nanoparticles in water or ethylene glycol depicted. Fig. 2. Schematic of the problem nanofluid in a porous Channel and magnetic field. u + x y = 1 nf c p u T T + x y P nf y + nf 2 x y 2 2 T = k nf x + 2 T 2 y 2 u 2 + nf 4 y The suitable boundary conditions for the velocity are: 3 2. PROBLEM DESCRIPTION Steady two-dimensional laminar flow of an incompressible viscous electrically conducting fluid in a channel with permeable walls with a long rectangular plate with uniform translation in x,l x over an infinite porous plate and made to consider. The distance between the two plates is h. We are observing a normal velocity q on the porous walls. A uniform magnetic field B is assumed to be applied towards direction y Figs In the case of a short circuit to neglect the electrical field and perturbations to the basic normal field and without any gravity forces, the governing equations are: 9 u x + v u u u + x y = 1 P nf x + nf y = u x + 2 u 2 y 2 u nf B 2 nf 2 y = 0 u = u 0 = q T = T 0 5 y = h u = u 0 = q T = T h 6 Calculating a mean velocity U by the relation: U h = h 0 u dy = L x q 7 We consider the following transformations: x = x L x y = y h u u= U = q P y = P q 2 = T T 0 T h T 0 Then, we can consider four dimensionless numbers: the Hartmann number Ha for the description of magnetic forces 11 and the Reynolds number Re for dynamic forces, Prandtl number and Eckert number: f Ha = Bh 9 f f 8 Re = hq nf 10 Table I. Different models for simulation of Dynamic viscosity. Model Electrical conductivity Dynamic viscosity I nf 3 s / f 1 = 1 + f s / f + 2 s / f 1 nf = f Fig. 1. Three dimensional Schematic of the problem. II nf 3 s / f 1 = 1 + f s / f + 2 s f 1 nf = f J. Adv. Phys., 3, 1 11, 2014

3 Fakou et al. Investigation of Nanofluid MHD Flow and Heat Transfer in a Channel Table II. Thermo physical properties of nanofluids and nanoparticles. Density Electrical Specific heat Thermal Material conductivity capacity C P conductivity K Silver Copper Ethylene glycol Drinking water Ec = Pr = nf C P k nf 11 U 2 C p T h T 0 12 where the effective density nf and specific heat capacity C P are defined as: 12 nf = f 1 + s C P nf = C p f 1 + C p s 13 The effective thermal conductivity of the nanofluid can be approximated by the Maxwell Garnett s MG model as: 13 k nf = k s + 2k f 2 k f k s 14 k f k s + 2k f + k f k s where is the nanoparticle volume fraction. Different models for simulating dynamic viscosity of the nanofluids are shown in Table I. In the first model, the effective thermal conductivity and viscosity of nanofluid are calculated by the Maxwell 13 and Brinkman models, Table I. The thermo physical properties of the nanofluid are given in Table II. Introducing Eqs into Eqs. 1 4 leads to the dimensionless equations: u x + y = 0 15 u u u + P x y = 2 y x + nf hp u Ha2 Re 2 2 u x + 2 u 2 y 2 B A 16 Fig. 3. Comparison of HPM and numerical results for dimensionless velocities and temperature a U y,bv y and c y. J. Adv. Phys., 3, 1 11,

4 Investigation of Nanofluid MHD Flow and Heat Transfer in a Channel Fakou et al. u + x C D Pr y = p y y + nf hq [ Re u + x y 2 = 2 x y x y 2 u 2 ] y Ec where A, B, C and D are constant parameters: A = 1 + s f 3 B s / f 1 = 1 + s / f + 2 s / f 1 C = k s + 2k f 2 k f k s k s + 2k f + k f k s D = 1 + C P s C P f Quantity of is defined as the aspect ratio between distance h and a characteristic length L x of the slider. This ratio is normally small. Berman s similarity transformation is used to be free from the aspect ratio of : = V y u= u U = u 0 U y + x dv dy 21 Introducing Eq. 21 in the second momentum Eqs. 19, 20 shows that quantity P y / y does not depend on the longitudinal variable x. With the first momentum equation, we also observe that 2 P y / x 2 is independent of x. Table IV. Comparison between y results from applied method for Re = Ha = Pr = Ec = 1and = Y HPM NUM Error We omit asterisks for simplicity. Then a separation of variables leads to: 11 UV VU = 1 1 Re A U Ha 2 B U 22 V IV = Ha 2 B V + ReA V V VV 23 Table III. Comparison between V y and U y results from applied method for Re = Ha = 1and = Y HPM U HPM V NUM U NUM V Error U Error J. Adv. Phys., 3, 1 11, 2014

5 Fakou et al. Investigation of Nanofluid MHD Flow and Heat Transfer in a Channel = C D Pr Re V +Ec u 0 U 2 24 Where primes denote differentiation with respect to y and asterisks have been omitted for simplicity. The dynamic boundary conditions are: y = 0 U= 1 V = 1 V = 0 = 0 y = 1 U= 1 V = 1 V = 0 = 1 25 where A u is defined as follows: A u = L u + N u 28 Homotopy-perturbation structure is shown as: H p = L L u 0 + p L u 0 + p N f r = 0 29 or 3. DESCRIBE HOMOTOPY PERTURBATION METHOD AND APPLIED TO THE PROBLEM 3.1. Describe Homotopy Perturbation Method Consider the following function: A u f r = 0 26 with the boundary condition of: B = u u = 0 27 n H p = 1 p L L u 0 + p A f r = 0 30 where r p 0 1 R 31 Obviously, considering Eqs. 29 and 30, we have: H 0 = L L u 0 = 0 32 H 1 = A f r = 0 Where p 0 1 is an embedding parameter and u 0 is the first approximation that satisfies the boundary condition. Fig. 4. Effect of nanoparticle volume fraction,, onav y,bu y and c y, for water with copper nanoparticles when Re = 1andHa= 1. J. Adv. Phys., 3, 1 11,

6 Investigation of Nanofluid MHD Flow and Heat Transfer in a Channel Fakou et al. The process of the changes in p from zero to unity is that of v r p changing from u 0 to u r. We consider v as: = 0 + p 1 + p and the best approximation is: u = lim p 1 = the above convergence is discussed in Refs. [28 30] The HPM Applied to the Problem A homotopy perturbation method can be constructed as follows: H U p = 1 p 1 Re 1 U A p UV VU 1 Re A U Ha 2 B U 1 35 H V p = 1 p V IV + p V IV Ha 2 B V + ReA V V VV 36 H p = 1 p + p + C D Pr Re V + Ec u 0 U 2 37 One can now try to obtain a solution of Eqs in the form of: U y = U 0 y + pu 1 y + p 2 U 2 y + 38 V y = V 0 y + pv 1 y + p 2 V 2 y + 39 y = 0 y + p 1 y + p 2 2 y + 40 where v i Y i = are functions yet to be determined. According to Eqs the initial approximation to satisfy boundary condition is: U 0 Y = 1 41 V 0 Y = 4Y 3 + 6Y Y = Y 43 Fig. 5. Effect of Hartmann number Ha on dimensionless velocities and temperature for water with copper nanoparticles, = 0.04, a V y, Re = 1, b U y,re= 1, c y, Re= 1, d V y,re= 5, e U y,re= 5, f y, Re= 5. 6 J. Adv. Phys., 3, 1 11, 2014

7 Fakou et al. Investigation of Nanofluid MHD Flow and Heat Transfer in a Channel Substituting Eqs into Eqs yields: d2 dy U 1 Y 12Y Y = Y 3 + 6Y Y Y Y 24Y d4 dy 4 V 1 Y = y 3 + 6y d2 = 0 46 d 1 2 The solutions of Eqs may be written as follow: U 1 Y = 0 507Y Y Y Y 47 V 1 Y = 0 276Y Y Y Y Y Y Y Y = 0 661Y Y Y Y 49 In the same manner, the rest of components were obtained by using the Maple package. According to the HPM, we can conclude: U Y = Y Y Y Y 0 062Y Y Y Y 5 50 V Y = 0 655Y Y Y Y Y Y Y Y Y Y 3 51 Y = 0 424Y Y Y Y Y Y Y Y Y 2 52 and so on. In the same manner the rest of the components of the iteration formula can be obtained. 4. RESULTS AND DISCUSSION In the present study HPM method is applied to obtain an explicit analytic solution of the laminar nanofluid flow and Fig. 6. Effect of Reynolds number Re on dimensionless velocities and temperature for water with copper nanoparticles, = 0.04, a V y, Ha = 1, b U y,ha=1, c y, Ha= 1, d V y,ha= 10, e U y,ha= 10, f y, Ha= 10. J. Adv. Phys., 3, 1 11,

8 Investigation of Nanofluid MHD Flow and Heat Transfer in a Channel Fakou et al. Fig. 7. Effect of Prandtl and Eckert number on dimensionless temperature for water with copper nanoparticles, = 0.04, a y, Ha= Re = Ec = 1, b y, Ha= Re = Ec = 10, c y, Ha= Re = Pr = 1. Table V. U y variations for the different types of nanofluids and nanoparticles, Re = Ha = 1and = Y Water-copper Water-silver Ethylene glycol-copper Ethylene glycol-silver J. Adv. Phys., 3, 1 11, 2014

9 Fakou et al. Investigation of Nanofluid MHD Flow and Heat Transfer in a Channel Table VI. V y variations for the different types of nanofluids and nanoparticles, Re = Ha = 1and = Y Water-copper Water-silver Ethylene glycol-copper Ethylene glycol-silver heat transfer in a channel with permeable walls in the presence of uniform magnetic field Fig 1. First, a comparison between the applied method, HPM and Numerical method is investigated. For this aim Eqs are solved for different nanofluid structures see Table II and comparison between the described methods is demonstrated in Figure 3. Other comparisons are presented in Tables III and IV. It can be concluded from table values that HPM is very close to numerically method. Figure 4 shows the effect of nanoparticle volume fraction on U y, V y and y for water with copper nanoparticles when Re = Ha = Pr = Ec = 1. Nanoparticle volume fraction has no significant effect on velocity and temperature profiles. Effect of Hartmann number Ha on dimensionless velocities and temperature for water with copper is shown in Figure 5. Applied magnetic field, the channels will not have much impact on the V y and y profiles. But generally increasing Hartmann number decreases rapidly, especially in the middle of the channel and this decreases rapidly is Table VII. y variations for the different types of nanofluids and nanoparticles, Re = Ha = Pr = Ec = 1and = Y Water-copper Water-silver Ethylene glycol-copper Ethylene glycol-silver J. Adv. Phys., 3, 1 11,

10 Investigation of Nanofluid MHD Flow and Heat Transfer in a Channel Fakou et al. more sensible in high Reynolds number. The maximum value of U y in the middle of the channel when the Hartmann and Reynolds number is small, it happens. Effect of Reynolds number Re on dimensionless velocities and temperature is shown in Figure 6. When the Hartmann number is small, the effect of Reynolds number on the velocities and temperature profiles are more sensible. Generally decreases rapidly with increasing Reynolds number and the maximum value of U y in the middle of the channel when the Hartmann and Reynolds number is very small, it happens. when Re = 0 the temperature profile is linear and with increasing Reynolds number the temperature profile is parabolic and with increasing Reynolds number, decreases temperature and heat transfer will be done later. Effect of Prandtl and Eckert number on dimensionless temperature is shown in Figure 7. When Hartmann and Reynolds number is small, the effect of Prandtl number on the temperature profile is more sensible. When the Pr = 0, the temperature profile is linear and with the Prandtl number increases profile is parabolic and temperature decreases. Eckert number variation has little impact on the temperature profile. The effects of the nanoparticle and liquid phase material on velocities and temperature profiles are shown in Tables V VII for U y, V y and y, respectively. These tables reveal that when nanofluid includes copper as nanoparticles or ethylene glycol as fluid phase in its structure, the U y, V y and y values are greater than the other structures. 5. CONCLUSION In this study, Homotopy Perturbation Method is applied to solve the problem of laminar nanofluid flow and heat transfer in a channel with permeable walls in the presence of uniform magnetic field. First a comparison between the applied method, HPM and Numerical method is investigated. The results show Homotopy perturbation method is very close to numerically method. In general, increasing the Reynolds number is increases the nanofluid flow velocity in the channel and the maximum amount of temperature increase and by applied magnetic field Ha, velocity and the maximum amount of temperature in the channel is reduces. Increasing Prandtl number and Eckert, respectively, the maximum value of theta increases and decreases. Also nanoparticle volume fraction has no significant effect on velocity and temperature profiles but the effect in the velocity profile is minimal when volume fraction of nanoparticle is decreased. NOMENCLATURE HPM Homotopy perturbation method NUM Numerical method P Pressure q Mass transfer parameter Re Reynolds number U Dimensionless velocity in the x direction V Dimensionless velocity in the y direction h Suspension height Ec Eckert number Ha Hartmann number Pr Prandtl number Length of the slider L x C p Specific heat k Thermal conductivity T = T h T 0 Difference temperature between the plates Fluid thermal diffusivity u 0 x velocity of the pad u Dimensionless x-component velocity v Dimensionless y-component velocity Velocity component in the x direction Velocity component in the y direction x Dimensionless horizontal coordinate y Dimensionless vertical coordinate x Distance in the x direction parallel to the plates y Distance in the y direction parallel to the plates. u v Greek Symbols Fluid density Dimensionless temperature Kinematic viscosity Electrical conductivity Aspect ratio h/lx. References and Notes 1. V. Wernert, O. Schäf, H. Ghobarkar, and R. Denoyel, Micro Porous Mesoporous Mater 83, A. Jafari, P. Zamankhan, S. M. Mousavi, and P. Kolari, Commun. Nonlinear Sci. Numer. Simul. 14, A. R. Goerke, J. Leung, and S. R. Wickramasinghe, Chem. Eng. Sci. 57, S. S. Mneina and G. O. Martens, AEU Int. J. Electron. Commun 63, Y. H. Andoh and B. Lips, Appl. Therm. Eng. 23, A. Runstedtler, Chem.Eng.Sci.61, A. S. Berman, J. Appl. Phys. 24, A. K. S. P. Chandran and N. C. Sacheti, Int. Commun. Heat Mass Transf. 23, M. Sheikholeslami, D. Domairry, H. R. Ashorynejad, and I. Hashim, Sains Malaysiana 10, S. Soleimani, M. Sheikholeslami, D. D. Ganji, and M. Gorji-Bandpy, Int. Commun. Heat Mass Transf. 39, M. Sheikholeslami, H. R. Ashorynejad, G. Domairry, and I. Hashim, Hindawi Publ. Corp. J. Appl. Math. 2012, M. A. A. Hamad, I. Pop, and A. I. Md Ismail, Nonlinear Anal. Real World Appl. 12, D. Domairry, H. R. Sheikholeslami, M. Ashorynejad, R. Subba, R. Gorla, and M. Khani, Nanoeng. Nanosyst. 225, M. Sheikholeslami, M. Gorji-Bandpy, and D. D. Ganji, Int. Commun. Heat Mass Transf. 39, J. Adv. Phys., 3, 1 11, 2014

11 Fakou et al. Investigation of Nanofluid MHD Flow and Heat Transfer in a Channel 15. M. Sheikholeslami, M. Gorji-Bandpy, D. D. Ganji, S. Soleimani, and S. M. Seyyedi, Int. Commun. Heat Mass Transf. 39, M. Sheikholeslami, D. D. Ganji, H. R. Ashorynejad, and H. B. Rokni, Appl. Math. Mech. English Ed. 1, M. Hatami and D. D. Ganji, J. Mol. Liq. 188, M. Hatami, R. Nouri, and D. D. Ganji, J. Mol. Liq. 187, M. Hatami and D. D. Ganji, J. Mol. Liq. 190, G. Domairry and M. Hatami, J. Mol. Liq. 193, M. Hatami and D. D. Ganji, Case Studies in Thermal Engineering 2, M. Hatami and D. D. Ganji, Energy Convers. Manage. 78, M. Hatami, M. Sheikholeslami, and D. D. Ganji, J. Mol. Liq. 195, M. Hatami, M. Sheikholeslami, M. Hosseini, and D. D. Ganji, J. Mol. Liq. 194, M. Sheikholeslami, M. Hatami, and D. D. Ganji, J. Mol. Liq. 194, H. R. Ashorynejad, A. A. Mohamad, and M. Sheikholeslami, Int. J. Therm. Sci. 64, A. Desseaux, Int. J. Eng. Sci. 37, J. C. Maxwell, A Treatise on Electricity and Magnetism, Second edn,, Oxford University Press, Cambridge 1904, pp M. Fakour, A. Vahabzadeh, D. D. Ganji, and I. Rahimi Petroudi, Cent. Eur. J. Eng. 4, D. D. Ganji and A. Sadighi, J. Comput. Appl. Math. 207, M. Fakour, A. Vahabzadeh, and D. D. Ganji, International Journal of Case Studies in Thermal Engineering 2014, M. Sheikholeslami, M. Hatami, and D. D. Ganji, Powder Technol. 246, M. Fakour, D. D. Ganji, A. Vahabzadeh, and S. H. H. Kachapi, Walailak Journal of Science and Technology 11, J. Adv. Phys., 3, 1 11,

ABSTRACT. Keywords: Laminar viscous flow; magnetic field; optimal homotopy asymptotic method; porous channel ABSTRAK

ABSTRACT. Keywords: Laminar viscous flow; magnetic field; optimal homotopy asymptotic method; porous channel ABSTRAK Sains Malaysiana 41(1)(212): 1281 1285 Investigation of the Laminar Viscous Flow in a Semi-Porous Channel in the Presence of Uniform Magnetic Field using Optimal Homotopy Asymptotic Method (Kajian tentang