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30120140502004 2

  1. 1. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – INTERNATIONAL JOURNAL OF MECHANICAL ENGINEERING 6340(Print), ISSN 0976 – 6359(Online), Volume 5, Issue 2, February (2014), pp. 26-35, © IAEME AND TECHNOLOGY (IJMET) ISSN 0976 – 6340 (Print) ISSN 0976 – 6359 (Online) Volume 5, Issue 2, February (2014), pp. 26-35 © IAEME: www.iaeme.com/ijmet.asp Journal Impact Factor (2014): 3.8231 (Calculated by GISI) www.jifactor.com IJMET ©IAEME EFFECT OF PARTICLE SIZE AND CHEMICAL REACTION ON CONVECTIVE HEAT AND MASS TRANSFER OF A MHD NANO FLUID IN A CYLINDRICAL ANNULUS *1G.V.P.N. SRIKANTH, 4 2 Dr. G. SRINIVAS, 3 B. SURESH BABU, Dr. B. TULASI LAKSHMI DEVI 1,2,4 3 Department of Mathematics, Guru Nanak Institute of Technology, Hyderabad, A.P, India Department of Mathematics, Sreyas Institute of Engineering & Technology, Hyderabad, A.P, India. ABSTRACT The present work deals with the effect of size of the nano-particle and the liquid like layer formed duo to the natural chemical reaction of the liquid with the metical particle. The particle size and the layer around the particle certainly alter the heat and mass transfer. We have study the effects of particle size and the layer on convective heat and mass transfer of MHD nano-fluid with heat source and chemical reaction when the nano-fluid flows in a cylindrical annulus filled with porous metrical. The coupled governing equations are solved using method of lines in Mathematical package. Key words: MHD Nano - Fluid, Annulus, Porous Material, Chemical Reaction. 1. INTRODUCTION The investigation for enhancing the heat and mass transfer through a fluid has given birth to the nano-fluid duo to the presents of metical particle in the fluid enhances the heat and mass transfer widely. The enough literature is providing an excellent support for the enhancement of heat and mass transfer. Xuan and Li [10] presented a study on the thermal conductivity of a nano-fluid consisting of copper nano-particles. The measured data showed that adding 2.5-7.5% copper oxide nano particles to the water increases its conductivity by about 24-78%. Rafee [4] has concluded that for a constant heat transfer rate, by increasing the length ratio of the annulus (L/Dh), firstly the entropy generation ratio will decrease until reaches its minimum value. 26
  2. 2. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print), ISSN 0976 – 6359(Online), Volume 5, Issue 2, February (2014), pp. 26-35, © IAEME Free convection flow and heat transfer in hydro magnetic case is important in nuclear and space technology (Singh KR 1963[6], Yu CP 1969[8], Yu CP 1970[9]). In particular, such convection flow in a vertical annulus region in the presence of radial magnetic field has been studied by Sastry and Bhadram (1978)[5].Nanda and Purushotham (1976)[2] have analyzed the free convection of a thermal conducting viscous incompressible fluid induced by traveling thermal waves on the circumference of a long vertical circular cylindrical pipe. Whitehead (1972)[7], Neeraja (1993)[3] has made a study of the fluid flow and heat transfer in a viscous incompressible fluid confined in an annulus bounded by two rigid cylinders. The flow is generated by periodical traveling waves imposed on the outer cylinder and the inner cylinder is maintained at constant temperature. Recently Ali J. Chamkha [1] has studied the Heat and Mass Transfer from MHD Flow over a Moving Permeable Cylinder with Heat Generation or Absorption and destructive Chemical Reaction. He found that the diffusion decreases with increase in chemical reaction. The above studies motivated to investigate theoretically the effects of particle size, chemical reaction and heat source on convective heat and mass transfer in a nano - fluid while passing through horizontal porous annulus in the presence of magnetic field. 2. FORMULATION OF THE PROBLEM We consider free and force convection flow of Cu-water nano-fluid through a porous medium in a circular cylindrical annulus in the presence of heat source, whose inner and outer walls are maintained at a constant temperature also the concentration is constant on the both walls. The flow velocity, temperature and concentration in the fluid to be fully developed. The fluid region has constant physical properties and the flow is a mixed convection flow taking place under thermal and molecular buoyancies and uniform axial pressure gradient. The boussenissque approximation is invoked so that the density variation is confined to the thermal and molecular buoyancy forces. In the momentum, energy and diffusion are coupled and non-linear. Also the flow in is unidirectional along the axial cylindrical annulus. Making use of the above assumptions the governing equations are ∂ (r u) = 0 ∂r (1) g ( ρ β T ) (T − T0 ) g ( ρ βc ) ( c − c0 ) µ n f  ∂2 w 1 ∂ w ∂2 w  µ n f σ B02 nf nf + + 2 − w− w+ + = 0  2 ρ nf  ∂r ρ nf ρ nf ρ nf r ∂r ∂ z  ρ nf k (2) α ∂T ∂z w D nf ∂  ∂c  ∂c = r  + kl ( c − c0 ) ∂z r ∂r  ∂r  = nf r ∂ ∂r  ∂T  r −  ∂r  w Q (T − T0 ) ( ρ cp ) n f (3) (4) 27
  3. 3. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print), ISSN 0976 – 6359(Online), Volume 5, Issue 2, February (2014), pp. 26-35, © IAEME The appropriate initial and boundary conditions for the problem are given by T = T0 , c = c0 , w = 0 on r = a (5) T = Tm , c = cm , w = 0 on r = a + s Thermo-Physical properties are related as follows: k ρ = (1 − φ ) ρ n f n f (ρ c ) p n f +φρ , α = f s n f (ρ c ) = (1 − φ ) ( ρ c ) + φ ( ρ c ) p nf p f p s (ρ β ) µ nf where = (1 − φ ) ( ρ β ) f + φ (ρ β ) s     = 1 + 2.5 φ + 4.5    h d  p  n f µ k nf f = k 1     2 + h  1 + h   d  d  p  p   2           d f 1− φ ) + β k φ + c k Re 2 d pr φ ( f 1 p 1 d p p p (6) β = 0.01 is a cons tan t for considering the kapitza resis tan ce per unit area 1 c = 18 × 106 is a proportionality cons tan t 1 d Re d p = γ p f κT 3π µ d l f p f = 1.381×1023 T γ 3π µ ( 0.738) f f ν d l f f = 0.384 nm for water , pr = Pr andtl number = = Mean free path = 0.738 , κ = α f f Boltzman n cons tan t , T = 300 k 28
  4. 4. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print), ISSN 0976 – 6359(Online), Volume 5, Issue 2, February (2014), pp. 26-35, © IAEME The thermo-physical properties (values) of the materials used are as follows. Table 1: Physical Properties Water Copper(Cu) ρ ( kg / m ) κ (W / m K ) 4,179 997.1 0.613 385 8,933 400 β ×10−5 (1/ K) 21 1.67 β × 106 (m2/ h) 298.2 3.05 µ 8.94 × 10 −4 ----- C p ( J / kg K ) 3 T c We introduce the following dimensionless variables: R= T − T0 c − c0 r z w u , Z = , W = , U = , θ = ,C = νf νf a a Tm − T0 cm − c0 (7) Using equations 5, 6, 7 the Eqs. 2, 3 & 4 can be written in the following dimensionless form:               ∂2 W 1 ∂ W  1 1 1 + 2.5 φ + 4.5    2 +  2   ∂ R R ∂R       1− φ + φ   h  2 + h  1 + h        d  d   d    p  p      p  1 −  1− φ + φ   W ∂θ ∂z ∂C ∂z ρs   ρf   MW + =     θ     1  ∂2 C 1 ∂ C  +   + KC S c  ∂ R2 R ∂R 29 1 W  ρs  1− φ + φ   ρf    1 1 Gr θ + Gc θ = 0   ρs  ρs  1− φ + φ  1− φ + φ    ρf  ρf         ρ 2c k k 1  f pf p p − 52 = 1 − φ + 0.01 φ + φ 28632.9991 ×10 2 3 Pr  k k d µ  f  f p f       1 1 − QH  (ρ c ) Pr  p s 1 − φ + φ (ρ c )  p f  W            1 1 − 1 + 2.5 φ + 4.5   2 ρs  D          h  2 + h  1 + h    ρf     d  d   d    p  p      p          1   (ρ c )   p s  1 − φ + φ  (ρ c )   p f     ∂2 θ 1 ∂θ   ∂ R2 + R ∂ R   
  5. 5. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print), ISSN 0976 – 6359(Online), Volume 5, Issue 2, February (2014), pp. 26-35, © IAEME Where the corresponding boundary conditions (5) can be written in the dimensionless form as: W = 0 , θ = 0 , c = 0 on R = 1 W = 0 , θ = 1, c = 1 on R = 1 + s Here pr is the Prandtl number, M is the magnetic parameter(Hartmann number), QH is the heat source parameter, S c is the Schmidt number, K is the chemical Reaction parameter, D −1 is the Darcy number, G r is the Grashof number, G c is the Molecular Grashof number, which are defined as: νf νf σ B0 2 k Q 1 a a , QH = a , Sc = = Pr = , M = , K = l a, kf Dn f D k αf µf νf Gr = g (ρ β T )( T m − T0 ) ν 2 , Gc = f g (ρ β c )( cm − c0 ) ν 2 f The local Nusselt number Nu in dimension less form: k nf Nu = − θ ' (1 + s ) kf 3. SOLUTION OF THE PROBLEM The cross section of the cylinder has considered which appears in the annulus form for numerical computations. The governing equations are solved for momentum (w), temperature ( θ ) and diffusion (C) by using method of lines with the help of Mathematica package across the cylindrical annulus subject to the boundary conditions. 4. RESULTS The profiles of momentum, temperature and diffusion are drawn at Pr = 7 and for constant axial temperature and axial concentration gradients. From Figs. 1 - 11 the flow is maximum in the mid region of the annulus. The momentum decreases with solid volume fraction ( φ ).If the amount of Cu nano-particles increases the momentum decreases due to Brownian motion. The width of the annulus affects the flow very much. The flow is maximum for maximum width of the annulus (s). The velocity enhances with increase of the thickness of the layer (h) around the nano-particle due to increase in friction. The velocity is maximum for 10nm or more and almost constant for 2nm or less thickness. The flow has obstructed by the Cu nano-particle very much. As the Particle size (dp) increases from 20nm to 100nm the velocity is decreased due to low Brownian motion. The velocity is maximum for small particles at about 20nm and the velocity is almost constant at about 100nm or more. The velocity increases with increase of both molecular (Gc) and thermal Grashof numbers (Gr). The velocity decreases with increase in the Hartmann number (M) and the velocity found maximum in the absence of the magnetic field. The velocity decreases with increase in heat source (QH). The velocity increases with increase in Schmidt number (Sc). The velocity increases with increase in chemical reaction parameter. The velocity is more for destructive (K > 0) chemical reaction than the generative (K < 0) chemical reaction. The flow increases with increase in the porosity ( D −1 ). 30
  6. 6. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print), ISSN 0976 – 6359(Online), Volume 5, Issue 2, February (2014), pp. 26-35, © IAEME From Figs. 12 – 18 the heat transfer takes place from inner cylinder to outer cylinder linearly for all variations of φ, s, h, dp, QH, K and D-1. The temperature enhances with increase in volume fraction ( φ ) of Cu particles. Temperature increases rapidly with increase of the width (s) of the annulus. The temperature is maximum in the annulus when the inner and outer cylinders are of same radius. The layer (h) around the nano-particle reduces the temperature due to less thermal conductivity of CuOH when compared with Cu. The temperature enhances with increase in size of the particle (dp). The temperature decreases with increase in heat source (QH). The temperature is more when the chemical reaction is generative (K<0) and less when the chemical reaction is destructive (K>0). The temperature reduces with the increase in porosity ( D −1 ). From Figs. 19 – 22, the diffusion is linear for all variations of φ, s, Sc and K. The diffusion increases with increase of φ, s, Sc and K. the Diffusion is rapidly increasing with the increase in the width (s) of the annulus and is very high for same size of inner and outer cylinders of annulus. The diffusion more during the destructive (K>0) chemical reaction and the diffusion is less during the generative (K<0) chemical reaction with the Cu-particle. 9.00E-03 2.50E-02 8.00E-03 2.00E-02 7.00E-03 6.00E-03 1.50E-02 5.00E-03 4.00E-03 1.00E-02 φ = 0,0.02,0.04,0.05 s = 0.2, 0.5, 0.8, 1 3.00E-03 5.00E-03 2.00E-03 1.00E-03 0.00E+00 0.00E+00 1 2 3 4 5 6 7 8 9 101112131415161718192021 -1.00E-03 1 1.05 1.1 1.15 1.2 1.25 1.3 1.35 1.4 1.45 1.5 -5.00E-03 Fig.1 Variation of w with φ Fig.2 Variation of w with s 0.025 0.018 0.016 0.02 0.014 0.015 0.012 h = 2,4,6,8,10 0.01 0.01 dp = 20,40,60,80,100 0.008 0.006 0.005 0.004 0.002 0 0 -0.002 1 1 1.05 1.1 1.2 1.25 1.3 1.35 1.4 1.45 1.5 -0.005 Fig.3 Variation of w with h 1.05 1.1 1.15 1.2 1.25 1.3 1.35 1.4 1.45 1.5 Fig.4 Variation of w with d p 31
  7. 7. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print), ISSN 0976 – 6359(Online), Volume 5, Issue 2, February (2014), pp. 26-35, © IAEME 0.012 0.012 0.01 0.01 0.008 0.008 Gc = 2,5,8,10 0.006 0.006 0.004 0.004 0.002 Gr = 2,5,8,10 0.002 0 0 1 1.05 1.1 1.15 1.2 1.25 1.3 1.35 1.4 1.45 1.5 1 1.051.1 1.15 1.2 1.251.3 1.35 1.4 1.45 1.5 -0.002 -0.002 Fig.5 Variation of w with G c Fig.6 Variation of w with G r 0.009 0.009 0.008 0.008 0.007 0.007 0.006 0.006 0.005 0.005 0.004 M = 0,5,10 0.004 0.003 0.002 0.002 0.001 QH = 2,5,8,10 0.003 0.001 0 0 -0.001 1 1.05 1.1 1.15 1.2 1.25 1.3 1.35 1.4 1.45 1.5 -0.001 1 1.05 1.1 1.15 1.2 1.25 1.3 1.35 1.4 1.45 1.5 Fig.8 Variation of w with QH Fig.7 Variation of w with M 0.009 0.009 0.008 0.008 0.007 0.007 0.006 0.006 0.005 Sc = 0.3,0.6,1.3 0.005 0.004 0.004 0.003 0.003 0.002 0.002 0.001 K = -0.5,-0.2,0,0.2,0.5 0.001 0 -0.001 1 1.051.11.151.21.251.31.351.41.451.5 0 -0.001 1 1.051.1 1.151.2 1.251.3 1.351.4 1.451.5 Fig.9 Variation of w with Sc Fig.10 Variation of w with K 32
  8. 8. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print), ISSN 0976 – 6359(Online), Volume 5, Issue 2, February (2014), pp. 26-35, © IAEME 0.009 1 0.008 0.9 0.007 0.8 0.006 0.7 0.005 0.6 0.5 0.004 1/D = 2,5,8,10 φ = 0,0.02,0.04,0.05 0.4 0.003 0.3 0.002 0.2 0.001 0.1 0 0 -0.001 1 1.05 1.1 1.15 1.2 1.25 1.3 1.35 1.4 1.45 1.5 1 1.05 1.1 1.15 1.2 1.25 1.3 1.35 1.4 1.45 1.5 Fig.11 Variation of w with D−1 Fig.12 Variation of θ with φ 1 1 0.9 0.9 0.8 0.8 0.7 0.7 s = 0.2,0.5,0.8,1 0.6 0.6 0.5 0.5 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1 h = 2,4,6,8,10 0 0 1 1.05 1.1 1.15 1.2 1.25 1.3 1.35 1.4 1.45 1.5 1 2 3 4 5 6 7 8 9 101112131415161718192021 Fig.13 Variation of θ with s Fig.14 Variation of θ with h 1 1 0.9 0.9 0.8 0.7 0.8 0.7 dp = 20,40,60,80,100 0.6 0.6 0.5 0.5 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1 0 QH = 2,5,8,10 0 1 1.05 1.1 1.15 1.2 1.25 1.3 1.35 1.4 1.45 1.5 1 1.05 1.1 1.15 1.2 1.25 1.3 1.35 1.4 1.45 1.5 Fig.15 Variation of θ with d p Fig.16 Variation of θ with QH 33
  9. 9. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print), ISSN 0976 – 6359(Online), Volume 5, Issue 2, February (2014), pp. 26-35, © IAEME 1 1 0.9 0.9 0.8 0.8 0.7 0.7 0.6 0.6 1 / D = 2,5,8,10 0.5 0.5 K = -0.5,-0.2,0,0.2,0.5 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1 0 0 1 1.05 1.1 1.15 1.2 1.25 1.3 1.35 1.4 1.45 1.5 1 1.05 1.1 1.15 1.2 1.25 1.3 1.35 1.4 1.45 1.5 Fig.18 Variation of θ with D−1 Fig.17 Variation of θ with K 1 1 0.9 0.9 0.8 0.8 0.7 0.6 s = 0.2,0.5,0.8,1 0.7 φ = 0,0.02,0.04,0.05 0.6 0.5 0.5 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1 0 0 1 1.05 1.1 1.15 1.2 1.25 1.3 1.35 1.4 1.45 1.5 1 2 3 4 5 6 7 8 9 101112131415161718192021 Fig.19 Variation of c with φ Fig.20 Variation of c with s 1 1 0.9 0.9 0.8 0.8 0.7 0.6 0.7 Sc = 0.3,0.6,1.6 K = -0.5,-0.2,0,0.2,0.5 0.6 0.5 0.5 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1 0 0 1 1.05 1.1 1.15 1.2 1.25 1.3 1.35 1.4 1.45 1.5 1 1.05 1.1 1.15 1.2 1.25 1.3 1.35 1.4 1.45 1.5 Fig.21 Variation of c with Sc Fig.22 Variation of c with K 34
  10. 10. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print), ISSN 0976 – 6359(Online), Volume 5, Issue 2, February (2014), pp. 26-35, © IAEME Nusselt Number: Table-2 s = 0.5, h = 4, G c = 5, G r = 5, M = 5, S c = 0.6, D −1 = 5 K = 0.2, d p = 40, K = 0.2, d p = 40, K = 0.2, d p = 40, K = 0.2, d p = 20, K = 0.2, d p = 100, K = − 0.2, d p = 40, K = 0, d p = 40, QH = 0 QH = 5 QH = 10 QH = 5 QH = 5 QH = 5 QH = 5 0.02 -1.64838 -2.35051 -2.96791 -2.35512 -2.34821 -2.35049 -2.3505 0.04 -1.64798 -2.29046 -2.86112 -2.29467 -2.28836 -2.29045 -2.29046 0.05 -1.64781 -2.26412 -2.81397 -2.26814 -2.2621 -2.26411 -2.26411 ߶ The variation of Nusselt number for different values of volume fraction, K, dp and QH are depicted in the Table-2 on the surface of the outer cylinder. The magnitude of heat transfer decreases as the volume fraction increases on the surface of the outer cylinder. The magnitude of heat transfer increases as the heat source increases. The magnitude of heat transfer decreases with increase in particle size. The magnitude of heat transfer increases from generative to destructive chemical reaction. REFERENCES [1] Ali. J. Chamkha, Heat and Mass Transfer from MHD Flow over a Moving Permeable Cylinder with Heat Generation or Absorption and Chemical Reaction. Communications in Numerical Analysis, 2011 doi:10.5899/2011. [2] Nanda RS and Prushotham R (1976) : Int.dedication seminar on recent advances on maths and applications, Vaeanasi. [3] Neeraja,G(1993):Ph.D thesis, S.P.Mahila University, Tirupathi, India. [4] R. Rafee,Entropy Generation Calculation for Laminar Fully Developed Forced Flow and Heat Transfer of Nanofluids inside Annuli, Journal of Heat and Mass Transfer Research xxx (2013). [5] Sastri VUK and Bhadram CVV(1978) : App,Sci.Res,V.34,2/3.p.117 [6] Singh KR and Cowling TJ (1963) : Quart.J.Maths.Appl.Maths,V.16.p.1. [7] Whitehead JA (1972):Observations of rapid means flow produced mercury by a moving heater, Geophys. Fluid dynamics, V.3, pp.161-180. [8] Yu CP and Yong,H(1969): Appl.Sci.Res,V.20,p.16. [9] Yu CP (1970):Appl.Sci.Res, V.22, p.127. [10] Y.M. Xuan, Q. Li, Heat transfer enhancement of nanofluids, Int. J. Heat Fluid Flow, 21, 58–64, (2000). [11] Dr P.Ravinder Reddy, Dr K.Srihari and Dr S. Raji Reddy, “Combined Heat and Mass Transfer in MHD Three-Dimensional Porous Flow with Periodic Permeability & Heat Absorption”, International Journal of Mechanical Engineering & Technology (IJMET), Volume 3, Issue 2, 2012, pp. 573 - 593, ISSN Print: 0976 – 6340, ISSN Online: 0976 – 6359. [12] Kavitha T, Rajendran A, Durairajan A, Shanmugam A, “Heat Transfer Enhancement Using Nano Fluids And Innovative Methods - An Overview”, International Journal of Mechanical Engineering & Technology (IJMET), Volume 3, Issue 2, 2012, pp. 769 - 782, ISSN Print: 0976 – 6340, ISSN Online: 0976 – 6359. 35

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