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Entropy generation and energy conversion rate for the peristaltic flow in a tube with magnetic field
Energy xxx (2015) 1e8
Contents lists available at ScienceDirect
Energy journal homepage: www.elsevier.com/locate/energy
Entropy generation and energy conversion rate for the peristaltic flow in a tube with magnetic field Noreen Sher Akbar DBS&H, CEME, National University of Sciences and Technology, Islamabad, Pakistan
a r t i c l e i n f o
a b s t r a c t
Article history: Received 3 November 2014 Received in revised form 8 December 2014 Accepted 10 December 2014 Available online xxx
Impact of entropy generation for the peristaltic flow in a tube is investigated. The entropy generation number due to heat transfer and fluid friction is formulated. The velocity and temperature distributions across the tube are presented along with pressure attributes. Exact analytical solution for velocity and temperature profile is obtained. Velocity, temperature, pressure gradient, pressure rise, Bejan number and streamlines are presented for radius of the tube a, Hartmann number M, amplitude ratio ε, Brinkman number Br and flow Q presented graphically. It is found that the entropy generation number attains high values in the region close to the walls of the tube, while it gains low values near the center of the tube. © 2014 Elsevier Ltd. All rights reserved.
Keywords: Entropy generation Heat transfer Viscous fluid Uniform tube
1. Introduction Heat Transfer and fluid flow processes are inherently irreversible, which leads to an increase in entropy generation and thus, destruction of useful energy. The optimal second law design criteria depend on the minimization of entropy generation encountered in fluid flow and heat transfer processes. Bejan [1] showed that entropy generation in convective fluid flow is due to heat transfer and viscous shear stresses. Numerical studies on the entropy generation in convective heat transfer problems were carried out by different researchers. Drost and White [2] developed a numerical solution procedure for predicting local entropy generation for a fluid impinging on a heated wall. Further detail analysis on entropy is presented by Benedetti and Sciubba [3] they presented numerical calculation of the local rate of entropy generation in the flow around a heated finned tube. Abu-Hijleh et al. [4] studied numerical prediction of entropy generation due to natural convection from a horizontal cylinder. In another article numerical prediction of entropy generation in separated flows is discussed by Abu-Nada [5]. Slip law effects on heat transfer and entropy generation of pressure driven flow of a power law fluid in a microchannel under uniform heat flux boundary condition is presented by Anand [6]. Guelpa et al. [7] discussed entropy generation analysis for the design
E-mail addresses: [email protected], [email protected].
improvement of a latent heat storage system. They show that the improved system allows to reduce PCM solidification time and increase second law efficiency. Second law analysis of heat transfer in laminar flow for hexagonal cross-section duct was analyzed analytically Oztop et al. [8]. An analysis of entropy generation through circular duct with different shaped longitudinal fins of laminar flow is discussed by Dagtekin et al. [9]. Further analysis on entropy generation can be seen through Refs. [10e13]. Peristalsis is a radially symmetrical contraction and relaxation of muscles which propagates in a wave down a muscular tube, in an anterograde fashion. Peristalsis is often found in the contraction of smooth muscle tissue to propel food/chyme through a digestive tract, such as the human gastrointestinal tract, after the pioneer work done by Latham [14], this mechanism become an interesting topic for the researchers, A new numerical solution for the MHD peristaltic flow of a bio-fluid with variable viscosity in a circular cylindrical tube via Adomian decomposition method is presented by Ebaid [15]. Nadeem and Akbar [16] discussed influence of heat and chemical reactions on the peristaltic flow of a Johnson Segalman fluid in a vertical asymmetric channel with induced MHD. Akbar and Nadeem [17] discussed convective heat transfer of a Sutterby fluid in an inclined asymmetric channel with partial slip. Ellahi et al. [18]. studied series solutions of Magnetohydrodynamic peristaltic flow of a Jeffrey fluid in eccentric cylinders. Maraj et al. [19] analyze biological analysis of Jeffrey nano fluid in a curved channel with heat dissipation. Very recently Akbar [20] studied peristaltic flow of Tangent Hyperbolic fluid with convective
http://dx.doi.org/10.1016/j.energy.2014.12.034 0360-5442/© 2014 Elsevier Ltd. All rights reserved.
Please cite this article in press as: Akbar NS, Entropy generation and energy conversion rate for the peristaltic flow in a tube with magnetic field, Energy (2015), http://dx.doi.org/10.1016/j.energy.2014.12.034
2
N.S. Akbar / Energy xxx (2015) 1e8
vu vu vp v vu 2 vu u r u þw ¼ þm 2 þm vr vr vr vr vr r vr r v vu vw þ ; þm vz vr vz
Nomenclature a b c
l u w
r m cp
q k Br M P ε B0 Pr Ec
radius of the tube amplitude of the sinusoidal wave wave speed wavelength velocity in the r direction velocity in the z direction density viscosity specific heat temperature thermal conductivity of the fluid Brinkmann number Hartmann number pressure amplitude ratio applied magnetic field Prandtl number Eckert number
vw vw vp v vw 1 v vu vw r u þw ¼ þm 2 þm þ r vr vz vz vz vz vz vr r vr sB2o ðw þ cÞ; (5)
2. Problem formulation Let us consider the peristaltic flow of an incompressible, natural convective peristaltic flow of in a horizontal uniform tube (walls of the tube are smooth i.e. radius of the tube is same in all section of the tube). Sinusoidal wave is propagating along the walls of the tube. We choose a cylindrical coordinate system ðR; ZÞ , where Z-axis lies along the center line of the tubes and R axis is normal to it. Wave is propagating with a velocity c along the wall of the tube. Keeping in view the analysis geometry of the wall surface is defined as
2p Z ct ; l
rcp
v
vT vT þw vr vz
!
" v2 T
# 1 vT v2 T þ 2 r vr vz vr ! 2 2 ! vu vw vu vw 2 þ þ þ : þm 2 vz vr vr vz
¼k
þ 2
(6)
boundary condition. Further recent literature related to the topic includes Refs. [17,21e29]. The entropy generation for the peristaltic flow is not discussed so far up to yet, to fill this gap entropy generation for the peristaltic flow in a tube is investigated in the present article. The entropy generation number due to heat transfer and fluid friction is formulated. The velocity and temperature distributions across the tube are presented along with pressure attributes. Exact analytical solution for velocity and temperature profile is obtained. It is found that the entropy generation number attains high values in the region close to the walls of the tube, while it gains low values near the center of the tube.
h¼ aþb sin
(4)
where r and z are the coordinates. z is taken as the center line of the tube and r transverse to it, u and w are the velocity components in the r and z directions respectively, T is the local temperature of the fluid. Further, r is the effective density, m is the effective dynamic viscosity, (rcp) is the heat capacitance and k is the effective thermal conductivity of the fluid (Fig. 1).
T T0 r z w lu a2 p a ct ;d ¼ ;q ¼ r ¼ ;z ¼ ;w ¼ ;u ¼ ;p ¼ ;t ¼ ; a l c ac clm l l T0 M2 ¼
sB20 a2 h ; Br ¼ Ec Pr; h ¼ : a m (7)
with the help of Eq. (7), Eqs. (3)e(6) can be written as follows
dp ¼0 dr
(8)
dp 1 v vw ¼ r M2 ðw þ 1Þ ; dz r vr vr
(9)
2 1 v vq vw r þ Br ¼ 0; r vr vr vr
(10)
The non-dimensionless boundary conditions are defined as follows
(1)
In the fixed coordinates system ðR; ZÞ; flow between the two tubes is unsteady. It becomes steady in a wave frame ðr; zÞ moving with the same speed as the wave moves in the Z direction. The transformations between the two frames are:
r ¼ R; z ¼ Z ct; v ¼ V; w ¼ W c; p z; r; t ¼ p Z; R; t
(2)
The governing equations for the flow of an incompressible fluid can be written as:
1 vðruÞ vw þ ¼ 0; vz r vr
(3)
Fig. 1. Geometry of the problem.
Please cite this article in press as: Akbar NS, Entropy generation and energy conversion rate for the peristaltic flow in a tube with magnetic field, Energy (2015), http://dx.doi.org/10.1016/j.energy.2014.12.034
N.S. Akbar / Energy xxx (2015) 1e8
vw vq ¼ 0; ¼ 0; vr vr
at r ¼ 0;
w ¼ 1; q ¼ 0;
at r ¼ h;
(11a)
Ns ¼ NH þ NB ;
(11b)
In order to see whether the one term is dominant to other a criterion namely irreversibility ratio, which is the ratio between entropy generation due to fluid friction and Joule dissipation to the total entropy generation due to heat transfer. The irreversibility ratio x is defined as:
3. Entropy generation analysis
Entropy generation can be defined as follows [ 17,16]
0
N x¼ B ¼ NH
!2 1 2 2 ! k @ vT vT A m vu vw þ SG ¼ þ þ 2 vr vz vz vr T0 T0 2 ! vu vw sB2 þ þ þ o ðw þ cÞ; vr vz T0 !2
(12) Dimensionless form of the Entropy generation with the help of Eq. (7) due to fluid friction and magnetic field is given as:
Ns ¼
SG ¼ SG0
vq 2 Br vw 2 M2 Br ðw þ 1Þ2 ; þ þ vr L vr L
(13)
The dimensionless form of SG is known as entropy generation number NS which is the ratio of actual entropy generation rate to the reference volumetric entropy generation SG0, which is defined as follows
SG0 ¼
kDT DT ;L ¼ ; T 0 a2 T0
(14)
The total entropy generation in Eq. (14) can be written as the sum
Ns ¼ NH þ NF þ NM ¼ NH þ NB ;
3
(15)
where NH is the entropy generation due to heat transfer, NF is the local entropy generation due to fluid friction irreversibility and NM is the entropy generation due to magnetic field caused by the motion of electrically conducting fluids under the magnetic field inducing electrical currents that circulates in the fluid. Further, the sum of second and third term in Eq. (16) can be treated as the entropy generation NB due to combined effects of fluid friction and magnetic field.
Br L
(16)
2
vw vr
2
þ MLBr ðw þ 1Þ2
2
;
(17)
vq vr
Note that heat transfer irreversibility dominates in the range 0 < x < 1 whereas x > 1 indicates that irreversibility is solely due to the sum of fluid friction and magnetic field irreversibility, The contribution of heat transfer to entropy generation is same to the combined effects of fluid friction and magnetic field when x ¼ 1. Alternatively, another irreversibility parameter is the Bejan number which is the ratio of heat transfer irreversibility to the total irreversibility due to heat transfer, fluid friction and magnetic field. Mathematically,
Be ¼
NH 1 ¼ 1þx NS
(18)
From Eq. (19), it is clear that Bejan number ranges from 0 to 1. Here Be / 0 is the limiting case when the entropy generation due to combined effects of fluid friction and magnetic field is dominant, Be / 1 is the opposite limit where heat transfer irreversibility dominates and Be ¼ ½ is the case where the contributions of both heat and fluid friction to entropy generation are equal.
4. Exact solutions Exact solutions have been evaluated for velocity, temperature and pressure gradient from Eqs. (9)e(11). Velocity of the fluid flow is
wðr; zÞ ¼ 1
dp 1 dp I0 ðMrÞ þ ; dz M 2 dz M2 I0 ðhMÞ
(19)
The temperature distribution is given as:
Fig. 2. Velocity profile w(r, z) against the radial distance r for different values of Hartmann number M and flow rate Q.
Please cite this article in press as: Akbar NS, Entropy generation and energy conversion rate for the peristaltic flow in a tube with magnetic field, Energy (2015), http://dx.doi.org/10.1016/j.energy.2014.12.034
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N.S. Akbar / Energy xxx (2015) 1e8
Fig. 3. Pressure gradient dp/dz versus z for different values of Hartmann number M and amplitude ratio ε.
Fig. 4. Pressure rise DP against the flow rate Q for different values of Hartmann number M and amplitude ratio ε.
Fig. 5. Temperature profile q(r, z) for different values of Brinkman number Br and Hartmann number M.
Please cite this article in press as: Akbar NS, Entropy generation and energy conversion rate for the peristaltic flow in a tube with magnetic field, Energy (2015), http://dx.doi.org/10.1016/j.energy.2014.12.034
N.S. Akbar / Energy xxx (2015) 1e8
dp 2 2 2 M h I1 ðhMÞ2 r 2 I1 ðMrÞ2 dz þ M2 r 2 1 I0 ðMrÞ2 MrI0 ðMrÞI1 ðMrÞ ! 2 2 dp 2 2 2 dp 4 þ Br þ 2M Br h M þ I0 ðhMÞ dz dz !, 2 dp I0 ðhMÞI1 ðhMÞ 2M 4 I0 ðhMÞ2 : þ Br hM dz
ZhðzÞ
qðr; zÞ ¼ Br
Flow rate is given by:
5
F ¼ 2p
rwdr;
(21)
0
this implies that
M 2 F þ ph2 I0 ðhMÞ dp ¼ ; dz ph2 I2 ðhMÞ (20)
(22)
where the mean flow rate Q is given as:
Fig. 6. Entropy NS against the radial distance r for different values of Brinkman number Br, Hartmann number M, temperature difference L, flow rate Q and amplitude ratio ε.
Please cite this article in press as: Akbar NS, Entropy generation and energy conversion rate for the peristaltic flow in a tube with magnetic field, Energy (2015), http://dx.doi.org/10.1016/j.energy.2014.12.034
6
N.S. Akbar / Energy xxx (2015) 1e8
1 ε2 1þ : Q ¼Fþ 2 2
5. Results and discussions
(23)
Integrating Eq. (22) over the interval [0, 1], we can find the pressure rise given by the expression:
Z1 DP ¼ 0
dp dz: dz
(24)
In this section, we present a brief graphical analysis of the exact analytical solutions of the governing problem. Fig. 2(a) and (b) represent the magnitude of the horizontal velocity of the fluid inside the tube. We see that with the increase in the Hartmann number M, i.e. ratio of electromagnetic force to the viscous force, the velocity decreases in the center of the tube and increases near the walls of the tube, while as we increase the flow rate Q, the magnitude of velocity takes a positive shift all around the tube. Also we note that the velocity attains its highest values in the center of
Fig. 7. Variation of Bejan number Be against the radial distance r for different values of Brinkman number Br, Hartmann number M, temperature difference L, flow rate Q, and amplitude ratio ε.
Please cite this article in press as: Akbar NS, Entropy generation and energy conversion rate for the peristaltic flow in a tube with magnetic field, Energy (2015), http://dx.doi.org/10.1016/j.energy.2014.12.034
N.S. Akbar / Energy xxx (2015) 1e8
7
Fig. 8. Streamlines for the variation of Hartmann number M.
Fig. 9. Streamlines for different values of flow rate.
the tube at r ¼ 0, while it sufficiently decreases at the walls of the tube. Fig. 3(a) and (b) depict that the pressure gradient certainly decreases with a decrease in the Hartmann number. However a non-uniform behavior is seen in case of ε. In that case pressure gradient seems to be directly proportional to ε for 0 z < 0.25 and inversely proportional for 0.25 < z < 0.75, this behavior of pressure gradient is oscillating. The pressure rise versus flow rate is shown in Fig. 4(a) and (b), from the graphical demonstration, it is visible that pressure rise is directly proportional to M, ε in the peristaltic pumping region and inversely proportional to the same in the augmented peristaltic region. Temperature of the fluid in the tube significantly increases with an increase in Brinkman number Br and a decrease in Hartmann number M. In comparison to the walls of the tube, higher temperature exists in the center where r ¼ 0. We note that with higher the values of the Brinkman number, i.e. the ratio of viscous heat generation to external heating, the lesser will be the conduction of heat produced by viscous dissipation and hence larger the temperature rise see Fig. 5(a,b). Fig. 6(a)e(e) are prepared to analyze the entropy generation with respect to change in different physical constraints involved. Fig. 6(a) and (b) depict that entropy generation is directly proportional to both Hartmann number and Brinkman number. It has larger values near the walls of the tube as compared to the center of the tube. It is to be noticed that for significantly larger values of these two parameters, entropy generation can be larger in the center of the tube than to those generated at the walls. A similar behavior is seen if we increase the flow rate Q. Nevertheless, the
higher the values of L, ε the smaller the entropy generation. Rapid change in this case is seen in the center of the tube see Fig. 6(c)e(e). Fig. 7(a)e(e) are prepared to analyze the Bejan number with respect to change in different physical constraints involved. Fig. 7(a)e(e) depict that with the increase in Hartmann number, Brinkman number, temperature difference, flow rate and amplitude ratio heat transfer irreversibility is high as compare to the total irreversibility due to heat transfer, fluid friction and magnetic field. It is also seen that the entropy generation structure Bejan number does not vary appreciably with the Hartmann number. This means that the balance between the entropy generation components i.e. the ratio of heat transfer irreversibility to the total irreversibility due to heat transfer, fluid friction and magnetic field has same influence due to magnetic field. Fig. 8. Shows streamlines for Hartmann number M. It is seen that the size of bolus decreases with the rise in M. Fig. 9 present the streamlines for the variation of flow rate Q. It is seen that with the variation of flow rate Q size of the bolus increases but number of bolus decreases.
References [1] Bejan A. A study of entropy generation in fundamental convective heat transfer. J Heat Transfer 1979;101:718e25. [2] Drost MK, White MD. Numerical prediction of local entropy generation in an impinging jet. J Heat Transfer 1991;113:823e9. [3] Benedetti PL, Sciubba E. Numerical calculation of the local rate of entropy generation in the flow around a heated finned tube. In: ASME, AES, 3; 1993.
Please cite this article in press as: Akbar NS, Entropy generation and energy conversion rate for the peristaltic flow in a tube with magnetic field, Energy (2015), http://dx.doi.org/10.1016/j.energy.2014.12.034
8
N.S. Akbar / Energy xxx (2015) 1e8
[4] Abu-Hijleh B, Abu-Qudais M, Abu-Nada E. Numerical prediction of entropy generation due to natural convection from a horizontal cylinder. Energy 1999;24:327e33. [5] Abu-Nada E. Numerical prediction of entropy generation in separated flows. Entropy 2005;7:234e52. [6] Anand V. Slip law effects on heat transfer and entropy generation of pressure driven flow of a power law fluid in a microchannel under uniform heat flux boundary condition. Energy 2014;76:716e32. [7] Guelpa E, Sciacovelli Adriano, Verda Vittorio. Entropy generation analysis for the design improvement of a latent heat storage system. Energy 2013;37: 128e38. [8] Oztop HF, Sahin AZ, Dagtekin I. Entropy generation through hexagonal crosssectional duct for constant wall temperature in laminar flow. Int J Energy Res 2004;28(8):725e37. [9] Dagtekin I, Oztop HF, Sahin AZ. An analysis of entropy generation through circular duct with different shaped longitudinal fins of laminar flow. Int J Heat Mass Transfer 2005;48(1):171e81. [10] Basak Tanmay, Anandalakshmi R, Kumar Pushpendra, Roy S. Entropy generation vs. energy flow due to natural convection in a trapezoidal cavity with isothermal and non-isothermal hot bottom wall. Energy 2012;37:514e32. [11] Dagtekin I, Oztop HF, Bahloul A. Entropy generation for natural convection in-Shaped enclosures. Int Comm Heat Mass Transfer 2007;34: 502e10. [12] Oztop HF, Al-Salem K. A review on entropy generation in natural and mixed convection heat transfer for energy systems. Renewable Sustain Energy Rev 2012;16(1):911e20. [13] Pakdemirli M, Yilbas BS. Entropy generation in a pipe due to non-Newtonian fluid flow: constant viscosity case. Sadhana 2006;31:21e9. [14] Latham TW. Fluid motion in a peristaltic pump. MS. Thesis. Cambridge: Massachusetts Institute of Technology; 1966. [15] Ebaid A. A new numerical solution for the MHD peristaltic flow of a bio-fluid with variable viscosity in a circular cylindrical tube via Adomian decomposition method. Phys Lett A 2008;372:5321e8. [16] Nadeem S, Akbar Noreen Sher. Influence of heat and chemical reactions on the peristaltic flow of a Johnson Segalman fluid in a vertical asymmetric channel with induced MHD. Taiwan Inst Chem Eng 2011;42:58e66.
[17] Akbar Noreen Sher, Nadeem S. Convective heat transfer of a sutterby fluid in an inclined asymmetric channel with partial slip. Heat Transfer Res 2014;45(3):219e40. [18] Ellahi R, Riaz A, Sohail S, Mushtaq M. Series solutions of magnetohydrodynamic peristaltic flow of a Jeffrey fluid in eccentric cylinders. J Appl Math Inf Sci 2013;7(4):1441e9. [19] Maraj EN, Akbar Noreen Sher, Nadeem S. Biological analysis of Jeffrey nano fluid in a curved channel with heat dissipation. IEEE Trans NanoBioscience 2014;13(4):1e7. [20] Akbar Noreen Sher. Peristaltic flow of tangent hyperbolic fluid with convective boundary condition. Eur Phys J Plus 2014;129:214. [21] Akbar Noreen Sher. Endoscopic effects on the peristaltic flow of Cu-water nanofluid. J Comput Theor Nanosci 2014;11. 1150e1155(6). [22] Akbar Noreen Sher. Peristaltic flow with Maxwell carbon nanotubes suspensions. J Comput Theor Nanosci 2014;11(7). 1642e1648(7). [23] Akbar Noreen Sher. Metallic nanoparticles analysis for the peristaltic flow in an asymmetric channel with MHD. IEEE Trans Nanotechnol 2014;13: 357e61. [24] Nadeem S, Maraj EN, Akbar Noreen Sher. Theoretical analysis for peristaltic flow of Carreau nano fluid in a curved channel with compliant walls. J Comput Theor Nanosci 2014;11:1443e52. [25] Akbar Noreen Sher. Heat and mass transfer effects on Carreau fluid model for blood flow through tapered arteries with stenosis. Int J Bio Math 2014;7: 1450004. [26] Akbar Noreen Sher. Double-diffusive natural convective peristaltic flow of a Jeffrey nanofluid in a porous channel. Heat Transfer Res 2014;45(4): 293e307. [27] Nadeem S, Maraj EN, Akbar Noreen Sher. Investigation of peristaltic flow of Williamson nano fluid in a curved channel with compliant walls. Appl Nanosci 2014;4:511e21. [28] Akbar Noreen Sher, Maraj EN, Nadeem S. Copper nanoparticle analysis for peristaltic flow in curved channel with heat transfer characteristics. Eur Phys J Plus 2014;129:149. [29] Akbar Noreen Sher, Butt Adil Wahid. CNT suspended nanofluid analysis in a flexible tube with ciliated walls. Eur Phys J Plus 2014;129:174.
Please cite this article in press as: Akbar NS, Entropy generation and energy conversion rate for the peristaltic flow in a tube with magnetic field, Energy (2015), http://dx.doi.org/10.1016/j.energy.2014.12.034
Contents lists available at ScienceDirect
Energy journal homepage: www.elsevier.com/locate/energy
Entropy generation and energy conversion rate for the peristaltic flow in a tube with magnetic field Noreen Sher Akbar DBS&H, CEME, National University of Sciences and Technology, Islamabad, Pakistan
a r t i c l e i n f o
a b s t r a c t
Article history: Received 3 November 2014 Received in revised form 8 December 2014 Accepted 10 December 2014 Available online xxx
Impact of entropy generation for the peristaltic flow in a tube is investigated. The entropy generation number due to heat transfer and fluid friction is formulated. The velocity and temperature distributions across the tube are presented along with pressure attributes. Exact analytical solution for velocity and temperature profile is obtained. Velocity, temperature, pressure gradient, pressure rise, Bejan number and streamlines are presented for radius of the tube a, Hartmann number M, amplitude ratio ε, Brinkman number Br and flow Q presented graphically. It is found that the entropy generation number attains high values in the region close to the walls of the tube, while it gains low values near the center of the tube. © 2014 Elsevier Ltd. All rights reserved.
Keywords: Entropy generation Heat transfer Viscous fluid Uniform tube
1. Introduction Heat Transfer and fluid flow processes are inherently irreversible, which leads to an increase in entropy generation and thus, destruction of useful energy. The optimal second law design criteria depend on the minimization of entropy generation encountered in fluid flow and heat transfer processes. Bejan [1] showed that entropy generation in convective fluid flow is due to heat transfer and viscous shear stresses. Numerical studies on the entropy generation in convective heat transfer problems were carried out by different researchers. Drost and White [2] developed a numerical solution procedure for predicting local entropy generation for a fluid impinging on a heated wall. Further detail analysis on entropy is presented by Benedetti and Sciubba [3] they presented numerical calculation of the local rate of entropy generation in the flow around a heated finned tube. Abu-Hijleh et al. [4] studied numerical prediction of entropy generation due to natural convection from a horizontal cylinder. In another article numerical prediction of entropy generation in separated flows is discussed by Abu-Nada [5]. Slip law effects on heat transfer and entropy generation of pressure driven flow of a power law fluid in a microchannel under uniform heat flux boundary condition is presented by Anand [6]. Guelpa et al. [7] discussed entropy generation analysis for the design
E-mail addresses: [email protected], [email protected].
improvement of a latent heat storage system. They show that the improved system allows to reduce PCM solidification time and increase second law efficiency. Second law analysis of heat transfer in laminar flow for hexagonal cross-section duct was analyzed analytically Oztop et al. [8]. An analysis of entropy generation through circular duct with different shaped longitudinal fins of laminar flow is discussed by Dagtekin et al. [9]. Further analysis on entropy generation can be seen through Refs. [10e13]. Peristalsis is a radially symmetrical contraction and relaxation of muscles which propagates in a wave down a muscular tube, in an anterograde fashion. Peristalsis is often found in the contraction of smooth muscle tissue to propel food/chyme through a digestive tract, such as the human gastrointestinal tract, after the pioneer work done by Latham [14], this mechanism become an interesting topic for the researchers, A new numerical solution for the MHD peristaltic flow of a bio-fluid with variable viscosity in a circular cylindrical tube via Adomian decomposition method is presented by Ebaid [15]. Nadeem and Akbar [16] discussed influence of heat and chemical reactions on the peristaltic flow of a Johnson Segalman fluid in a vertical asymmetric channel with induced MHD. Akbar and Nadeem [17] discussed convective heat transfer of a Sutterby fluid in an inclined asymmetric channel with partial slip. Ellahi et al. [18]. studied series solutions of Magnetohydrodynamic peristaltic flow of a Jeffrey fluid in eccentric cylinders. Maraj et al. [19] analyze biological analysis of Jeffrey nano fluid in a curved channel with heat dissipation. Very recently Akbar [20] studied peristaltic flow of Tangent Hyperbolic fluid with convective
http://dx.doi.org/10.1016/j.energy.2014.12.034 0360-5442/© 2014 Elsevier Ltd. All rights reserved.
Please cite this article in press as: Akbar NS, Entropy generation and energy conversion rate for the peristaltic flow in a tube with magnetic field, Energy (2015), http://dx.doi.org/10.1016/j.energy.2014.12.034
2
N.S. Akbar / Energy xxx (2015) 1e8
vu vu vp v vu 2 vu u r u þw ¼ þm 2 þm vr vr vr vr vr r vr r v vu vw þ ; þm vz vr vz
Nomenclature a b c
l u w
r m cp
q k Br M P ε B0 Pr Ec
radius of the tube amplitude of the sinusoidal wave wave speed wavelength velocity in the r direction velocity in the z direction density viscosity specific heat temperature thermal conductivity of the fluid Brinkmann number Hartmann number pressure amplitude ratio applied magnetic field Prandtl number Eckert number
vw vw vp v vw 1 v vu vw r u þw ¼ þm 2 þm þ r vr vz vz vz vz vz vr r vr sB2o ðw þ cÞ; (5)
2. Problem formulation Let us consider the peristaltic flow of an incompressible, natural convective peristaltic flow of in a horizontal uniform tube (walls of the tube are smooth i.e. radius of the tube is same in all section of the tube). Sinusoidal wave is propagating along the walls of the tube. We choose a cylindrical coordinate system ðR; ZÞ , where Z-axis lies along the center line of the tubes and R axis is normal to it. Wave is propagating with a velocity c along the wall of the tube. Keeping in view the analysis geometry of the wall surface is defined as
2p Z ct ; l
rcp
v
vT vT þw vr vz
!
" v2 T
# 1 vT v2 T þ 2 r vr vz vr ! 2 2 ! vu vw vu vw 2 þ þ þ : þm 2 vz vr vr vz
¼k
þ 2
(6)
boundary condition. Further recent literature related to the topic includes Refs. [17,21e29]. The entropy generation for the peristaltic flow is not discussed so far up to yet, to fill this gap entropy generation for the peristaltic flow in a tube is investigated in the present article. The entropy generation number due to heat transfer and fluid friction is formulated. The velocity and temperature distributions across the tube are presented along with pressure attributes. Exact analytical solution for velocity and temperature profile is obtained. It is found that the entropy generation number attains high values in the region close to the walls of the tube, while it gains low values near the center of the tube.
h¼ aþb sin
(4)
where r and z are the coordinates. z is taken as the center line of the tube and r transverse to it, u and w are the velocity components in the r and z directions respectively, T is the local temperature of the fluid. Further, r is the effective density, m is the effective dynamic viscosity, (rcp) is the heat capacitance and k is the effective thermal conductivity of the fluid (Fig. 1).
T T0 r z w lu a2 p a ct ;d ¼ ;q ¼ r ¼ ;z ¼ ;w ¼ ;u ¼ ;p ¼ ;t ¼ ; a l c ac clm l l T0 M2 ¼
sB20 a2 h ; Br ¼ Ec Pr; h ¼ : a m (7)
with the help of Eq. (7), Eqs. (3)e(6) can be written as follows
dp ¼0 dr
(8)
dp 1 v vw ¼ r M2 ðw þ 1Þ ; dz r vr vr
(9)
2 1 v vq vw r þ Br ¼ 0; r vr vr vr
(10)
The non-dimensionless boundary conditions are defined as follows
(1)
In the fixed coordinates system ðR; ZÞ; flow between the two tubes is unsteady. It becomes steady in a wave frame ðr; zÞ moving with the same speed as the wave moves in the Z direction. The transformations between the two frames are:
r ¼ R; z ¼ Z ct; v ¼ V; w ¼ W c; p z; r; t ¼ p Z; R; t
(2)
The governing equations for the flow of an incompressible fluid can be written as:
1 vðruÞ vw þ ¼ 0; vz r vr
(3)
Fig. 1. Geometry of the problem.
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vw vq ¼ 0; ¼ 0; vr vr
at r ¼ 0;
w ¼ 1; q ¼ 0;
at r ¼ h;
(11a)
Ns ¼ NH þ NB ;
(11b)
In order to see whether the one term is dominant to other a criterion namely irreversibility ratio, which is the ratio between entropy generation due to fluid friction and Joule dissipation to the total entropy generation due to heat transfer. The irreversibility ratio x is defined as:
3. Entropy generation analysis
Entropy generation can be defined as follows [ 17,16]
0
N x¼ B ¼ NH
!2 1 2 2 ! k @ vT vT A m vu vw þ SG ¼ þ þ 2 vr vz vz vr T0 T0 2 ! vu vw sB2 þ þ þ o ðw þ cÞ; vr vz T0 !2
(12) Dimensionless form of the Entropy generation with the help of Eq. (7) due to fluid friction and magnetic field is given as:
Ns ¼
SG ¼ SG0
vq 2 Br vw 2 M2 Br ðw þ 1Þ2 ; þ þ vr L vr L
(13)
The dimensionless form of SG is known as entropy generation number NS which is the ratio of actual entropy generation rate to the reference volumetric entropy generation SG0, which is defined as follows
SG0 ¼
kDT DT ;L ¼ ; T 0 a2 T0
(14)
The total entropy generation in Eq. (14) can be written as the sum
Ns ¼ NH þ NF þ NM ¼ NH þ NB ;
3
(15)
where NH is the entropy generation due to heat transfer, NF is the local entropy generation due to fluid friction irreversibility and NM is the entropy generation due to magnetic field caused by the motion of electrically conducting fluids under the magnetic field inducing electrical currents that circulates in the fluid. Further, the sum of second and third term in Eq. (16) can be treated as the entropy generation NB due to combined effects of fluid friction and magnetic field.
Br L
(16)
2
vw vr
2
þ MLBr ðw þ 1Þ2
2
;
(17)
vq vr
Note that heat transfer irreversibility dominates in the range 0 < x < 1 whereas x > 1 indicates that irreversibility is solely due to the sum of fluid friction and magnetic field irreversibility, The contribution of heat transfer to entropy generation is same to the combined effects of fluid friction and magnetic field when x ¼ 1. Alternatively, another irreversibility parameter is the Bejan number which is the ratio of heat transfer irreversibility to the total irreversibility due to heat transfer, fluid friction and magnetic field. Mathematically,
Be ¼
NH 1 ¼ 1þx NS
(18)
From Eq. (19), it is clear that Bejan number ranges from 0 to 1. Here Be / 0 is the limiting case when the entropy generation due to combined effects of fluid friction and magnetic field is dominant, Be / 1 is the opposite limit where heat transfer irreversibility dominates and Be ¼ ½ is the case where the contributions of both heat and fluid friction to entropy generation are equal.
4. Exact solutions Exact solutions have been evaluated for velocity, temperature and pressure gradient from Eqs. (9)e(11). Velocity of the fluid flow is
wðr; zÞ ¼ 1
dp 1 dp I0 ðMrÞ þ ; dz M 2 dz M2 I0 ðhMÞ
(19)
The temperature distribution is given as:
Fig. 2. Velocity profile w(r, z) against the radial distance r for different values of Hartmann number M and flow rate Q.
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Fig. 3. Pressure gradient dp/dz versus z for different values of Hartmann number M and amplitude ratio ε.
Fig. 4. Pressure rise DP against the flow rate Q for different values of Hartmann number M and amplitude ratio ε.
Fig. 5. Temperature profile q(r, z) for different values of Brinkman number Br and Hartmann number M.
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dp 2 2 2 M h I1 ðhMÞ2 r 2 I1 ðMrÞ2 dz þ M2 r 2 1 I0 ðMrÞ2 MrI0 ðMrÞI1 ðMrÞ ! 2 2 dp 2 2 2 dp 4 þ Br þ 2M Br h M þ I0 ðhMÞ dz dz !, 2 dp I0 ðhMÞI1 ðhMÞ 2M 4 I0 ðhMÞ2 : þ Br hM dz
ZhðzÞ
qðr; zÞ ¼ Br
Flow rate is given by:
5
F ¼ 2p
rwdr;
(21)
0
this implies that
M 2 F þ ph2 I0 ðhMÞ dp ¼ ; dz ph2 I2 ðhMÞ (20)
(22)
where the mean flow rate Q is given as:
Fig. 6. Entropy NS against the radial distance r for different values of Brinkman number Br, Hartmann number M, temperature difference L, flow rate Q and amplitude ratio ε.
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1 ε2 1þ : Q ¼Fþ 2 2
5. Results and discussions
(23)
Integrating Eq. (22) over the interval [0, 1], we can find the pressure rise given by the expression:
Z1 DP ¼ 0
dp dz: dz
(24)
In this section, we present a brief graphical analysis of the exact analytical solutions of the governing problem. Fig. 2(a) and (b) represent the magnitude of the horizontal velocity of the fluid inside the tube. We see that with the increase in the Hartmann number M, i.e. ratio of electromagnetic force to the viscous force, the velocity decreases in the center of the tube and increases near the walls of the tube, while as we increase the flow rate Q, the magnitude of velocity takes a positive shift all around the tube. Also we note that the velocity attains its highest values in the center of
Fig. 7. Variation of Bejan number Be against the radial distance r for different values of Brinkman number Br, Hartmann number M, temperature difference L, flow rate Q, and amplitude ratio ε.
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7
Fig. 8. Streamlines for the variation of Hartmann number M.
Fig. 9. Streamlines for different values of flow rate.
the tube at r ¼ 0, while it sufficiently decreases at the walls of the tube. Fig. 3(a) and (b) depict that the pressure gradient certainly decreases with a decrease in the Hartmann number. However a non-uniform behavior is seen in case of ε. In that case pressure gradient seems to be directly proportional to ε for 0 z < 0.25 and inversely proportional for 0.25 < z < 0.75, this behavior of pressure gradient is oscillating. The pressure rise versus flow rate is shown in Fig. 4(a) and (b), from the graphical demonstration, it is visible that pressure rise is directly proportional to M, ε in the peristaltic pumping region and inversely proportional to the same in the augmented peristaltic region. Temperature of the fluid in the tube significantly increases with an increase in Brinkman number Br and a decrease in Hartmann number M. In comparison to the walls of the tube, higher temperature exists in the center where r ¼ 0. We note that with higher the values of the Brinkman number, i.e. the ratio of viscous heat generation to external heating, the lesser will be the conduction of heat produced by viscous dissipation and hence larger the temperature rise see Fig. 5(a,b). Fig. 6(a)e(e) are prepared to analyze the entropy generation with respect to change in different physical constraints involved. Fig. 6(a) and (b) depict that entropy generation is directly proportional to both Hartmann number and Brinkman number. It has larger values near the walls of the tube as compared to the center of the tube. It is to be noticed that for significantly larger values of these two parameters, entropy generation can be larger in the center of the tube than to those generated at the walls. A similar behavior is seen if we increase the flow rate Q. Nevertheless, the
higher the values of L, ε the smaller the entropy generation. Rapid change in this case is seen in the center of the tube see Fig. 6(c)e(e). Fig. 7(a)e(e) are prepared to analyze the Bejan number with respect to change in different physical constraints involved. Fig. 7(a)e(e) depict that with the increase in Hartmann number, Brinkman number, temperature difference, flow rate and amplitude ratio heat transfer irreversibility is high as compare to the total irreversibility due to heat transfer, fluid friction and magnetic field. It is also seen that the entropy generation structure Bejan number does not vary appreciably with the Hartmann number. This means that the balance between the entropy generation components i.e. the ratio of heat transfer irreversibility to the total irreversibility due to heat transfer, fluid friction and magnetic field has same influence due to magnetic field. Fig. 8. Shows streamlines for Hartmann number M. It is seen that the size of bolus decreases with the rise in M. Fig. 9 present the streamlines for the variation of flow rate Q. It is seen that with the variation of flow rate Q size of the bolus increases but number of bolus decreases.
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[17] Akbar Noreen Sher, Nadeem S. Convective heat transfer of a sutterby fluid in an inclined asymmetric channel with partial slip. Heat Transfer Res 2014;45(3):219e40. [18] Ellahi R, Riaz A, Sohail S, Mushtaq M. Series solutions of magnetohydrodynamic peristaltic flow of a Jeffrey fluid in eccentric cylinders. J Appl Math Inf Sci 2013;7(4):1441e9. [19] Maraj EN, Akbar Noreen Sher, Nadeem S. Biological analysis of Jeffrey nano fluid in a curved channel with heat dissipation. IEEE Trans NanoBioscience 2014;13(4):1e7. [20] Akbar Noreen Sher. Peristaltic flow of tangent hyperbolic fluid with convective boundary condition. Eur Phys J Plus 2014;129:214. [21] Akbar Noreen Sher. Endoscopic effects on the peristaltic flow of Cu-water nanofluid. J Comput Theor Nanosci 2014;11. 1150e1155(6). [22] Akbar Noreen Sher. Peristaltic flow with Maxwell carbon nanotubes suspensions. J Comput Theor Nanosci 2014;11(7). 1642e1648(7). [23] Akbar Noreen Sher. Metallic nanoparticles analysis for the peristaltic flow in an asymmetric channel with MHD. IEEE Trans Nanotechnol 2014;13: 357e61. [24] Nadeem S, Maraj EN, Akbar Noreen Sher. Theoretical analysis for peristaltic flow of Carreau nano fluid in a curved channel with compliant walls. J Comput Theor Nanosci 2014;11:1443e52. [25] Akbar Noreen Sher. Heat and mass transfer effects on Carreau fluid model for blood flow through tapered arteries with stenosis. Int J Bio Math 2014;7: 1450004. [26] Akbar Noreen Sher. Double-diffusive natural convective peristaltic flow of a Jeffrey nanofluid in a porous channel. Heat Transfer Res 2014;45(4): 293e307. [27] Nadeem S, Maraj EN, Akbar Noreen Sher. Investigation of peristaltic flow of Williamson nano fluid in a curved channel with compliant walls. Appl Nanosci 2014;4:511e21. [28] Akbar Noreen Sher, Maraj EN, Nadeem S. Copper nanoparticle analysis for peristaltic flow in curved channel with heat transfer characteristics. Eur Phys J Plus 2014;129:149. [29] Akbar Noreen Sher, Butt Adil Wahid. CNT suspended nanofluid analysis in a flexible tube with ciliated walls. Eur Phys J Plus 2014;129:174.
Please cite this article in press as: Akbar NS, Entropy generation and energy conversion rate for the peristaltic flow in a tube with magnetic field, Energy (2015), http://dx.doi.org/10.1016/j.energy.2014.12.034