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Mixed convection–radiation interaction in a vertical porous channel: Entropy generation
Energy 28 (2003) 1557–1577 www.elsevier.com/locate/energy
Mixed convection–radiation interaction in a vertical porous channel: Entropy generation Shohel Mahmud , Roydon Andrew Fraser Department of Mechanical Engineering, University of Waterloo, 200 University Avenue West, Waterloo, On., Canada N2L 3G1 Received 30 October 2002
Abstract The present work examines analytically the effects of radiation heat transfer on magnetohydrodynamic mixed convection through a vertical channel packed with fluid saturated porous substances. First and Second Laws of thermodynamics are applied to analyze the problem. Special attention is given to entropy generation characteristics and their dependency on the various dimensionless parameters, i.e., Hartmann number (Ha), Plank number (Pl), Richardson number (Ri), group parameter (Br/II), etc. A steady-laminar flow of an incompressible-viscous fluid is assumed flowing through the channel with negligible inertia effect. The fluid is further considered as an optically thin gas and electrically conducting. Governing equations in Cartesian coordinates are solved analytically after reasonable simplifications. Expressions for velocity, temperature, local, and average entropy generation rates are analytically derived and presented graphically. # 2003 Elsevier Ltd. All rights reserved.
1. Introduction Any substance with a temperature above absolute zero transfers heat in the form of radiation. Thermal radiation always exits and can strongly interact with convection in many situations of engineering interest. The influence of radiation on natural or mixed convection is generally stronger than that on forced convection because of the inherent coupling between temperature and flow field (see [1]). Convection in a channel (or enclosed space) in the presence of thermal radiation continues to receive considerable attention because of its importance in many practical
Corresponding author. Tel.: +1-519-888-4567x3885; fax: +1-519-888-6197. E-mail address: [email protected] (S. Mahmud).
0360-5442/$ - see front matter # 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0360-5442(03)00154-3
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Nomenclature A B0 Be Br Br CK CP Eb Ec g G Gr Ha I k K n NF NH NS P P Pl Pr qR Re 000 S_ 000 Sc T u U v V w x X y Y
inverse Darcy number ¼ ve2 w=ðKU0 Þ magnetic induction (Wb m2) Bejan Bejan number ¼ NH =NS Brinkman number ¼ Ec Pr pffiffiffiffi modified Brinkman number ¼ Brw= K Forchheimer coefficient specific heat (kJ kg1 K1) blackbody radiation ¼ rT 4 Eckert number ¼ U02 =ðCP DTÞ gravitational acceleration (m s2) 0 a parameter ¼ qP =@X þ U 3 Grashof number ¼ ðgbDTw Þ=m2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Hartmann number ¼ B0 w re =ðqmÞ intensity of radiation thermal conductivity (W m1 K1) permeability (m2) an index ¼ 0 or 1 fluid friction irreversibility irreversibility due to heat transfer 000 000 entropy generation number ¼ S_ =Sc dimensionless pressure ¼ p=ðqf U02 Þ modified pressure ¼ P Re Plank number ¼ 16rT03 =k Prandtl number ¼ m=a radiative heat flux Reynolds number ¼ U 0 w=m entropy generation rate (W m3 K1) characteristic entropy transfer rate ¼ kDT 2 =ðw2 T02 Þ temperature ( C) fluid velocity in x direction (m s1) dimensionless axial velocity ¼ u=U 0 fluid velocity in y direction (m s1) dimensionless v velocity ¼ v=U 0 channel width (m) axial distance (m) dimensionless axial distance ¼ x=w transverse distance (m) dimensionless transverse distance ¼ y=w
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Greek a b e eL ; eR m q qf r re U X H j s P W C
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symbols thermal diffusivity (m2 s1) thermal expansion coefficient (K1) porosity emissivity kinematic viscosity (m2 s1) reflectivity density of fluid (kg m3) Stefan–Boltzmann constant fluid electrical conductivity (X1 m1) irreversibility distribution ratio ¼ N F =NS solid angle (st. rad.) dimensionless fluid temperature monochromatic absorption coefficient (m1) dimensionless optical thickness ¼ jw dimensionless temperature difference ¼ DT=T 0 porous-magnet parameter (see Eq. (21)) pffiffiffiffi porous media inertia coefficient ¼ CK e2 w= K
Subscript and superscript 0 reference value L value at left wall R value at right wall av average value
applications such as furnaces, combustion chambers, cooling towers, rocket engines, and solar collectors. During the past several decades, a number of experiments and numerical computations have been presented for describing the phenomenon of natural (or mixed) convection in channels or enclosures. These studies aimed at clarifying the effect of mixed convection on flow and temperature regimes arising from variations in the shape of the channel (or enclosure), in fluid properties, in the transition to turbulence, etc. Chawla and Chan [2] studied the effect of radiation heat transfer on thermally developing Poiseuille flow. The interaction of thermal radiation with conduction and convection in thermally developing, absorbing–emitting, nongray gas flow in a circular tube is investigated by Tabanfar and Modest [3]. Natural convection–radiation interaction is studied by Yucel et al. [4] for a square cavity, Lauriat [5] for a vertical cavity, and Chang et al. [6] for a complex enclosure. Akiyama and Chong [7] numerically analyzed the influence of gray surface radiation on the convection of nonparticipating fluid in a rectangular enclosed space. A combined free and forced convection flow of an electrically conducting fluid in a channel, in the presence of a magnetic field, is also of special technical significance because of its frequent occurrence in many industrial applications such as cooling of nuclear reactors, MHD marine
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propulsion, electronic packages, micro electronic devices, etc. Some other quite promising applications are in the field of metallurgy such as MHD stirring of molten metal and magnetic-levitation casting. Riley [8] studied buoyancy induced flow and transport in the presence of a magnetic field. For a rectangular vertical duct, Hunt [9] and Buhler [10] analyzed the fluid flow problem in a magnetic field with or without buoyancy effects. For a conducting fluid, Shercliff [11] analyzed the fluid flow characteristics in a pipe under a transverse magnetic field. Alboussiere et al. [12] did an asymptotic analysis to study buoyancy driven convection in a uniform magnetic field. Recently, Ghaly [13] and Chamkha et al. [14] analyzed the effect of radiation heat transfer on flow and thermal fields in the presence of a magnetic field for horizontal and inclined plates. References [15–18] presented some important aspects of magnetohydrodynamic free and/or mixed convection in a porous channel or enclosure. The foregoing discussions that deal with free, forced, and mixed convection problems of different geometries with or without radiation effect, in the presence and absence of a magnetic field in a porous or nonporous channel (or enclosure) are very much restricted to a First Law (of thermodynamics) analysis. The contemporary trend in the field of heat transfer and thermal design is to apply a Second Law (of thermodynamics) analysis, and its design-related concept of entropy generation minimization (EGM) (see [19]). This new trend is important and, at the same time, necessary, if the heat transfer community is to increase its contribution to viable engineering solutions to energy problems. Entropy generation is associated with thermodynamic irreversibility, which is present in all heat transfer processes. Because of the abundance of publications on entropy, no attempt is made here to review the literature; see Bejan [19] for a review on the application to problems of convection heat transfer and Arpaci [20] for radiation heat transfer. In this paper, we mainly focused on the entropy generation characteristics for a mixed convection flow inside a vertical porous channel with radiation heat transfer and under the action of transverse magnetic field.
2. Problem formulation Consider an optically thin and electrically conducting fluid flowing through a vertical channel packed with porous substance as shown in Fig. 1. The depth of the channel (along z-axis) is assumed sufficiently long compared to other dimensions. Both walls are isothermal and kept at the same or different temperatures. IL!R and IR!L represent the radiative intensity of the left and right walls, respectively. A transverse magnetic field of intensity B0 acts along the positive y-axis. Modeling the flow as ‘‘Boussinesq-incompressible’’ to take into account the coupling between the energy and momentum equations, we regard the density qf as constant everywhere except in the buoyancy term of momentum equation (Eq. (2)). The magnetic Reynolds number is assumed to be small so that the induced magnetic field is negligible, while the Hall effect of magnetohydrodynamics is assumed to be negligible. The flow through the porous medium is assumed to follow the Brinkman–Forchheimer model (see [21]) with additional body forces (buoyancy and magnetic forces). Correspondingly, the equations governing the steady-state con-
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Fig. 1. Schematic diagram of the problem under consideration.
servation of mass, momentum, and energy in the channel of Fig. 1 are @u @v þ ¼ 0; @x @y
me CK e 1 @u @u @2u 1 @p erB20 u; u þ pffiffiffiffi ujuj þ u þv ¼ egbðT T0 Þ þ m 2 K e @x @y @y qf @x qf K 2 @T @T @ T @2T 1 @qR þv ¼a ; u þ @x @y @x2 @y2 qf CP @y
(1) (2) (3)
where e, K, CK, b, re, B0, m, and a represent porosity, permeability, Forchheimer coefficient, volumetric thermal expansion coefficient, fluid electrical conductivity, magnetic flux, fluid kinematic viscosity, and thermal diffusivity, respectively. qR represents the radiative heat flux. A simplified approximation for calculating the radiative heat transfer term (qqR/@y) is presented in the next section before giving an approximate analytical solution to Eqs. (1)–(3).
3. Approximation of qqR/qy Most fluids of technological importance may be adequately described by the thin gas assumption. The behavior of thin gases near the boundary (solid vertical walls) is the knowledge we need for the present problem. Because of its negligible absorption, a thin gas is usually influenced by the geometry of the enclosures, and its behavior near a boundary depends on this
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geometry as well as on the boundary itself. A simplified mathematical model can be proposed for approximating qqR/@y term, according to Arpaci et al. [1], as follows: ð @qR (4) ¼ 4jEb j I dX; @y X where j, X, and Eb are the monochromatic absorption coefficient, solid angle, and blackbody emissive power, respectively. The second term at the right hand side of Eq. (4) represents the effect of boundaries. Assuming hemispherical isotropy, and replacing actual intensity with twostream intensities (outward and inward) as shown in Fig. 1, the second term of the right hand side of Eq. (4) can be written as ð I dX 2pðIL!R þ IR!L Þ: (5) X
Consider I0,L, I0,R are the intensities of radiation, eL ; eR are the emissivities, and qL, qR reflectivities for the left and the right walls, respectively. For a thin gas the absorption is negligible, and IL!R and IR!L remain uniform across the thickness of the gas between the plates (Fig. 1). Consequently IL!R and IR!L become identical. Approximating each stream as a combination of emission ðeI0 Þ and reflection (qI), IL!R and IR!L can be written as IL!R ¼ eL I0L þ qL IR!L
(6a)
IR!L ¼ eR I0R þ qR IL!R
(6b)
Now solutions to Eqs. (6a,b) for IL!R and IR!L yield IL!R ¼
eL I0L þ qL eR I0R ; 1 qL qR
IR!L ¼
eR I0R þ qR eL I0L : 1 qL qR
Substituting these intensities into Eq. (5) and transmissivity ¼ 0) and pI ¼ Eb , we get ð ð1=eR 1=2ÞEbL þ ð1=eL 1=2ÞEbR : I dX 4 ð1=eR 1=2Þ þ ð1=eL 1=2Þ X
(7) noting
that
qþe¼1
(assuming
(8)
Assuming that the emissivities of the left and right walls are identical ðeL ¼ eR Þ, substitution of Eq. (8) into Eq. (4) yields @qR 1 ¼ 4j Eb ðEbL þ EbR Þ : (9) 2 @y In the above expression, ðEbL þ EbR Þ=2 can be replaced by a mean emissive power Eb0. The two terms, Eb and Eb0, can be further expressed as rT4 and rT04 , where r is the Stefan– Boltzmann constant. The big difficulty arises due to the presence of the ‘‘T4’’ term in the energy equation, which essentially introduces a high nonlinearity and makes it impossible to obtain any analytical solution. Expressing T as a Taylor series about T0 with the assumption of small
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jTL TR j yields 1 X ðT T0 Þn dn 4 T ¼ f ðTÞ n! dT n T¼T0 n¼0 ¼ T04 þ 4ðT T0 ÞT03 þ
12ðT T0 Þ2 2 24ðT T0 Þ3 24ðT T0 Þ4 T0 þ T0 þ : 2 6 24
ð10Þ
Truncating the above series after the second term and using the definitions of Eb and Eb0, Eqs. (9) and (10), gives @qR
16jrT03 ðT T0 Þ: @y
(11)
4. Analysis The governing equations are put into their dimensionless forms with appropriate scaling parameters before giving velocity and temperature solutions. Lengths (x and y) are scaled with the channel width w. Velocity components (u and v) are scaled with the inlet velocity U0. The scaling factor for pressure is qU02 . Dimensionless temperature, H, is defined as ðT T0 Þ=DT, where T0 and DT are the reference temperature and reference temperature difference, respectively. For reference temperature T0, we chose ðTR þ TL Þ=2. For the reference temperature difference, DT, assuming that TR > T0 , we chose ðTR T0 Þ. The dimensionless forms of Eqs. (1)–(3) are then @U @V þ ¼ 0; @X @Y 1 @U @U @P 1 @2U @2U Gr Ha2 2 U þV þ U; þ H þ þ AU þ CU ¼ e @X @Y @X Re @X 2 @Y 2 Re2 Re 2 @H @H 1 @ H @2H Pl s U þV ¼ H; þ @X @Y RePr @X 2 @Y 2 RePr
(12) (13) (14)
where A, C, Re, Pr, Gr, Ha, and Pl represent inverse Darcy number, porous media inertia coefficient, Reynolds, Prandtl, Grashof, Hartmann, and Plank numbers, respectively. s is the dimensionless optical thickness. The definitions of the above parameters are given in the nomenclature section. The nonlinear nature of Eqs. (13) and (14) restricts us to solving them for a close, not exact, form of an analytical solution. Assume now that the flow is hydrodynamically and thermally fully developed (qU=qX ¼ 0, qH=qX ¼ 0). When combined with the continuity equation (Eq. (12)) and the impermeable boundary condition at the channel surface, we obtain V ¼ 0. 0 ) yields Expansion of U2 as a Taylor series about the dimensionless inlet velocity (U 1 X 0 Þn dn ðU U 2 2 þ 2ðU U 0 ÞU 0 þ ðU U 0 Þ2 : U ¼ f ðUÞ ¼U (15) 0 n n! dU 0 U¼U n¼0 Truncating the above series after the second term gives a reasonable approximation for U2 term in the momentum equation (last term on the left-hand side of Eq. (13)). After simplifi-
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cation, Eqs. (13) and (14) become pffiffiffiffiffiffiffiffiffiffiffi @2U @P 2 0 þ Ha2 U; GrRiH þ ARe þ 2CReU ¼ þ U 0 2 @Y @X @2H Pl sH ¼ 0; @Y 2
(16) (17)
where modified pressure ðP Þ is the product of dimensionless pressure (P) and Reynolds number (Re). Boundary conditions are At Y ¼ 0;
U ¼ 0 and H ¼ HL ; At Y ¼ 1;
U ¼ 0 and H ¼ HR :
(18)
Expressing the square root of the product of Plank number (Pl) and optical thickness (s) as the radiation parameter (Rd), the solution to Eq. (17) for the temperature H, subject to the thermal boundary conditions defined in Eq. (18), yields H ¼ C1 sinhðRd Y Þ þ C2 coshðRd Y Þ;
(19)
where C1 and C2 are two constants of integration and can be expressed by the following equation: 8 HR HL coshðRd Þ < ; C1 ¼ (20) sinhðRd Þ : C 2 ¼ HL : With the solution for H already determined, Eq. (16) can be solved for the velocity U subject to the no-slip boundary conditions given in Eq. (18). Omitting the detail arithmetic operations, the expression for the dimensionless velocity U becomes pffiffiffiffiffiffiffiffiffiffiffi C1 sinhðRd Y Þ þ C2 coshðRd Y Þ G (21) þ C3 sinhðWY Þ þ C4 coshðWY Þ 2 ; U ¼ GrRi 2 2 W W Rd where pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 þ Ha2 ; W ¼ ARe þ 2CReU
@P 2 G¼ þ U0 ; @X
where Richardson number (Ri) is the ratio between Grashof number (Gr) and square of the Reynolds number (Re2). Richardson number is an important parameter for characterizing mixed convective flows. It determines the relative dominance between buoyancy, viscous, and inertia forces. The constants C3 and C4 of Eq. (21) can be expressed as 1 G C1 sinhðRd Þ þ C2 coshðRd Þ C3 ¼ C coshðWÞ ; 4 sinhðWÞ W2 W2 R2d (22) G C2 : C4 ¼ 2 2 W W R2d In the above equation W is termed as a porous-magnet parameter. It is interesting to note that for equal magnitudes of W and Radiation parameter (Rd), the constants C3 and C4 are singular (1). This is an important restriction (W 6¼ Rd ) for the present problem. The magnitude
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of Reynolds number in mixed convection problems is usually kept low. For a porous medium with comparatively low porosity ðeÞ and high permeability (K), the parameter W reduces to the Hartmann number (see Eq. (21)). The dimensionless form of the average velocity along the X direction, Uav, can be evaluated by integrating Eq. (21) with respect to Y (0 Y 1), and yields pffiffiffiffiffiffiffiffiffiffiffi GrRi ½C CoshðWÞ þ C4 SinhðWÞ G
½C1 CoshðRd Þ þ C2 SinhðRd Þ þ 3 Uav ¼ 2 2 2 W W Rd W Rd pffiffiffiffiffiffiffiffiffiffiffi C3 C1 GrRi
: (23) W Rd W2 R2d
5. Second Law analysis The convection process in a channel is inherently irreversible. Nonequilibrium conditions arise due to the exchange of energy and momentum, within the fluid and at the solid boundaries. This causes continuous entropy generation. One part of this entropy production is due to the heat transfer in the direction of finite temperature gradients, which is common in all types of thermal engineering applications (see [19]). Another part of the entropy production arises due to the fluid friction and is generally termed as fluid friction irreversibility. Based on the assumptions already made, the volumetric rate of entropy generation for the present problem can be expressed as 2 2 _S000 ¼ k @T þn l @u þð1 nÞ l u2 : (24) 2 T0 @y KT0 T0 @y For a non-Darcy model of the porous media, the first two terms in Eq. (24) are used to calculate the local entropy generation rate (n ¼ 1). For a channel with a porous media that obeys Darcy’s model, entropy generation due to fluid friction can be evaluated using the third term (n ¼ 0) on the right hand side of Eq. (24). For details, see [19,22]. Entropy generation rate is positive and finite as long as temperature and velocity gradients are present in the medium. According to Bejan [19], the dimensionless form of the entropy generation rate is the entropy generation number (NS) which, by definition, equals the ratio of 000 actual entropy generation rate (S_ ) to the characteristic entropy transfer rate ðkDT 2 =ðw2 T02 ÞÞ. Using the same parameters, already used for scaling purpose, the dimensionless form of Eq. (24) is @H 2 Br @U 2 Br NS ¼ þn þð1 nÞ (25) ½U 2 : @Y P @Y P In Eq. (25), Br and P are Brinkman number and dimensionless temperature difference pffiffiffiffi (DT/ T0), respectively. The modified Brinkman number ðBr Þ is the product of Br and w= K . The ratio Br/P is commonly termed the group parameter. Combining Eqs. (19), (21) and (25), the
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dimensionless form of entropy generation number (NS) can be written as Br NS ¼ R2d ½C1 coshðRd Y Þ þ C2 sinhðRd Y Þ2 þn P " pffiffiffiffiffiffiffiffiffiffiffi #2 GrRi fC1 coshðRd Y Þ þ C2 sinhðRd Y Þg þ WfC3 coshðWY Þ þ C4 sinhðWY Þg W2 R2d þ ð1 nÞ
Br 2 U : P
ð26Þ
The first term at the right hand side of Eq. (26) represents the heat transfer part of entropy generation (NH) and the second part is the fluid friction contribution to entropy generation (NF).
6. Results and discussion It is difficult to study the influence of all parameters involved in the present problem on the flow and thermal field along with entropy generation characteristics. Therefore, a selected set of graphical results are presented in Figs. 2–14 that will provide a good understanding of the influence of different parameters on the velocity, temperature, and entropy generation profiles. Both symmetrical ðHL ¼ 1; HR ¼ 1Þ and asymmetrical temperatures ðHL ¼ 0; HR ¼ 1Þ at the walls are considered.
Fig. 2. Variation of velocity profile with radiation parameter.
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Fig. 3. Variation of velocity profile with porous-magnet parameter.
Fig. 4. Variation of temperature profile with radiation parameter.
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Fig. 5. Variation of entropy generation number with group parameter.
Fig. 6. Variation of entropy generation number with radiation parameter.
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Fig. 7. Variation of entropy generation number with porous-magnet parameter.
Fig. 8. Variation of entropy generation number with mixed convection parameter.
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Fig. 9. Average entropy generation as a function of radiation parameter.
Fig. 10. Variation of average entropy generation with mixed convection parameter at n ¼ 1.
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Fig. 11. Variation of average entropy generation with mixed convection parameter at n ¼ 0.
Fig. 12. Minimum entropy generation at different group parameters for n ¼ 1.
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Fig. 13. Minimum entropy generation at different group parameters for n ¼ 0.
Fig. 14. Rd,opt–(Gr Ri)0.5 relation at different group parameters for n ¼ 1.
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The roles of inverse Darcy number (A), porous media inertia coefficient (C), and Hartmann number (Ha) are very similar for the present problem. These parameters essentially add extra resistance to the flow field. Instead of analyzing individual influences, we analyze their influence as a group (influence of W). It should be noted that for a limiting case (low porosity and high permeability), the parameter W reduces to the Hartmann number (Ha). 6.1. Velocity distribution Modified dimensionless velocity ðU Þ is calculated by dividing the velocity U by the average velocity Uav. Fig. 2 shows the variation of U as a function of Y at different radiation parameters (Rd) associated with the case of constant and symmetrical wall temperatures. Velocity profiles are symmetrical about the centerline (Y ¼ 0:5) of the channel for each value of Rd. Increases in the value of Rd have a tendency to slow down the movement of the fluid in the middle portion of the channel. For Rd ¼ 6, the velocity profile is almost flat. For Rd ¼ 7 and 8, a maximum velocity occurs at two different locations (not at Y ¼ 0:5). Velocity gradient (qU/ qY) becomes zero at three different locations for Rd ¼ 7 and 8, which have zero contribution to entropy generation due to fluid flow (will discuss later in detail). The effect of the porous-magnet parameter (W) on the velocity profile is shown in Fig. 3 for symmetric temperatures ðHL ¼ HR ¼ 1Þ at the walls. The parameter W creates a resistive force similar to a drag force; it acts in the opposite direction of the fluid motion, thus causing the velocity of the fluid to decrease. This flow resistance is depicted by the decreases of the U as W increases in Fig. 3. 6.2. Temperature distribution Dimensionless temperatures (H) are plotted as a function of transverse distance Y in Fig. 4 for six selected values of radiation parameters. Temperatures at the walls are kept the same ðHL ¼ HR ¼ 1Þ. For Rd ¼ 0, Eq. (17) becomes a simple conduction equation and the corresponding temperature profile is similar to a conduction temperature profile. Temperature does not vary along the transverse distance and H ¼ 1 everywhere. For Rd > 0, the profiles become nonlinear but symmetric about the centerline of the channel. Hmin occurs at Y ¼ 0:5 and the magnitude of Hmin decreases with increasing Rd. For the symmetric temperature boundary condition ðHL ¼ HR Þ, the temperature gradient (qH/qY) is zero at the channel’s centerline where no heat transfer entropy is generated since there is no finite temperature gradient. 6.3. Idle point for entropy generation The location where the temperature gradient (qH/qY) is zero is an idle point for temperature entropy generation. No entropy generation occurs at points of zero temperature gradient. The location of such a point can be mathematically expressed by the following equation: 1 HR HL coshðRd Þ HL sinhðRd Þ ln : (27) Y j@H=@Y ¼0 ¼ 2Rd HR HL coshðRd Þ þ HL sinhðRd Þ
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It should be noted that, for a particular combination of HL, HR, and Rd, an idle point for temperature entropy may or may not exist. Any solution to Eq. (27) which yields an imaginary number, a value less than 0, or a value greater than 1, leads to a nonexisting idle point for temperature entropy. Similarly fluid friction irreversibility becomes zero at the location of zero velocity gradient (qU/qY). The complicated expression of velocity (see Eq. (21)) makes it difficult to derive an analytical expression (like Eq. (27)) for a fluid friction irreversibility idle point. An alternative approximate solution can be achieved by expanding the velocity gradient (qU/qY) in a series like @U X gC1 DRnd þ g C2 DRnd þ gC3 Wnþ1 þ g C4 Wnþ1 n ¼ n ¼ 0; 1; 2; . . . (28) Y ; @Y n! n where pffiffiffiffiffiffiffiffiffiffiffi Rd GrRi ; D¼ 2 W R2d
g ¼ 1 g;
1; when n is 0 or odd number; g ¼ 0; when n is even number
truncating the series after a desired order of n, set this truncated series to zero, and solving for Y ð¼ Y jqU=@Y ¼0 Þ. The accuracy of the solutions depends on n. 6.4. Entropy generation rate The foregoing discussions, which mainly focused on the flow field and thermal field characteristics in terms of velocity and temperature distribution profiles, are important because of their (velocity and temperature) close relation to entropy generation rate (see Eqs. (24) and (25)). Next we turn our attention to local and global entropy generation rates and EGM. Four parameters (Br/P, Rd, W, and (Gr Ri)0.5) are selected to show their influence on the local rate of entropy generation through graphical representation in Figs. 5–8. For each case, symmetric wall temperatures ðHL ¼ HR ¼ 1Þ are selected for the thermal boundary condition. Group parameter (Br/P) is an important dimensionless number for entropy generation analysis. It determines the relative importance of viscous effect to temperature entropy generation. Entropy generation number (NS) is plotted as a function of Y at different group parameters in Fig. 5. The entropy generation profile is symmetric about the centerline of the channel due to the symmetric distributions of velocity and temperature. For all group parameters, each wall acts as a strong concentrator of entropy generation because of the high near-wall gradients of velocity and temperature. For all Br/P, no entropy is generated at the centerline of the channel. Both velocity and temperature gradients (qU/@Y and qT/@Y) are zero at the centerline leaving no contribution to entropy generation. Due to high viscous effects, higher Br/P shows a larger NS at a particular location of Y. Fig. 6 presents the effect of radiation parameter (Rd) on local entropy generation rate. Entropy generation is characterized by concave shaped and symmetric profiles for all values of Rd. Small changes in Rd cause large variations in NS as shown in Fig. 6. Porous-magnet parameter (W) does not dominate like the radiation parameter. For small values of W, some variation is observed in the entropy generation profiles due to changes in W as shown in Fig. 7. This variation is almost negligible at higher values of W. Finally, in Fig. 8, the effect of mixed convection
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parameter ((Gr Ri)0.5) on entropy generation number is tested. A small range (0.1–5.0) of (Gr Ri)0.5 is selected. Entropy generation distribution is still characterized by a concave shaped profile. A small change in the value of (Gr Ri)0.5 causes a rapid rise in entropy generation. At a particular location of Y, NS increases with the increases in (Gr Ri)0.5. 6.5. Average entropy generation From the expression for entropy generation number (Eq. (26)), the volume averaged entropy generation rate can be evaluated using the following equation: ð 1 NS d8 ¼ NH;av þ NF;av ; (29) NS;av ¼ 8 8 where 8 is the volume of the channel. Average entropy generation number (NS,av) consists of two parts: the first part, NH,av, is the contribution of heat transfer, and the second part, NF,av, is the contribution of fluid friction. For a particular case, Fig. 9 shows the variation of average entropy generation rate with radiation parameter. Contributions to the total entropy generation are always an internal competition between irreversibilities due to heat transfer and fluid friction. For the selected range of Rd, NF,av decreases and NH,av increases with the increasing Rd. This opposite behavior of heat transfer and fluid friction irreversibilities with change in Rd leaves an optimum radiation parameter (Rd,opt) where total entropy generation in minimum. Fig. 10 shows the variation of NS,av with radiation parameter (Rd) at different mixed convection parameters (Gr Ri)0.5 and n ¼ 1. The reason behind choosing of parameters (Gr Ri)0.5 and Rd is due to their practical importance and strong contribution to entropy generation. For a particular (Gr Ri)0.5, NS,av decreases with increasing of Rd, and shows its minimum ([NS]min) at a particular value of Rd, then increases. For the selected range of (Gr Ri)0.5, NS,av profiles merge with each other after Rd 15. For n ¼ 0, Fig. 11 shows the distribution of NS,av with Rd while keeping other parameters the same as in Fig. 10. Profiles are similar to those in Fig. 10 except that the variation of magnitude of NS,av differs. 6.6. Entropy generation minimization The contemporary trend in the field of heat transfer and thermal design is Second Law (of thermodynamics) analysis and its design-related concept of entropy generation and its minimization ([19]). EGM is the method of modeling and optimization of real devices that owe their thermodynamic imperfection to heat transfer, mass transfer, and fluid flow irreversibilities. It is also known as thermodynamic optimization in engineering, where it was first developed, or more recently as finite space–time thermodynamics in the physics literature. The method combines from the start the most basic principles of thermodynamics, heat transfer, and fluid mechanics. Entropy generation minimization combines into simple models the most basic concepts of heat transfer, fluid mechanics, and thermodynamics. These simple models are used in the optimization of real (irreversible) devices and processes, subject to finite-size and finite-time constraints that are in fact responsible for the irreversible operation of the device. The combined heat transfer and thermodynamics model ‘‘visualizes’’ for the analyst the irreversible nature of the device. So, for proper use of this method, an analyst must know the behavior of system’s
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Fig. 15. Rd,opt–(Gr Ri)0.5 relation at different group parameters for n ¼ 0.
irreversibilities in terms of minimum entropy generation and the variation of irreversibilities with system parameters (flow parameters, transport properties, geometry, etc.). Figs. 12–15 present the important aspect of ‘‘entropy generation minimization (EGM)’’. The EGM criterion is the most appropriate measure of a real system’s approach to thermodynamic ideality (see [19]) and at the same time is a method of thermodynamic optimization for real systems. For n ¼ 1, [NS]min is plotted as a function of (Gr Ri)0.5 in Fig. 12 at different group parameters (Br/P). For all Br/P, [NS]min gradually increases with (Gr Ri)0.5. Fig. 13 shows the same distribution for n ¼ 0. The corresponding magnitude of the radiation parameter at which entropy becomes a minimum is termed Rd,opt, and which is plotted as a function of (Gr Ri)0.5 in Fig. 14 for n ¼ 1 and in Fig. 15 for n ¼ 0.
7. Conclusions We investigated analytically the First and Second Laws (of thermodynamics) aspects of fluid flow and heat transfer inside a vertical porous channel with a transverse magnetic field. Radiation heat transfer is considered assuming the fluid to be an optically thin gas. The influence of different dimensionless parameters, i.e. Rd, W, (Gr Ri)0.5, and Br/P, is tested on the calculated velocity (U), temperature (H), and local and average entropy generation number. Radiation parameter (Rd) has a strong influence on velocity and temperature. Higher values of Rd and W suppress the velocity profile around the centerline of the channel. The radiation parameter introduces nonlinearity in temperature profiles. An increase in the value of Rd shifts (in a non-
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linear fashion) the temperature profile away from the conduction profile (the linear one) which occurs at Rd ¼ 0. Entropy generation number is characterized by a concave shaped profile and is symmetric about the channel centerline for a symmetrical temperature boundary condition. Radiation and mixed convection parameters have a more dominating influence on entropy generation rate than porous-magnet and group parameters. Expressions for the idle points of entropy generation are derived and their location(s) are determined. Irreversibility due to the heat transfer and/or fluid friction become zero at these idle points. Based on entropy generation minimization, optimum radiation parameters (Rd,opt) are determined which increase with increasing (Gr Ri)0.5 and Br/P ðor Br =PÞ).
References [1] Arpaci VS, Selamet A, Kao SH. Introduction to heat transfer. New York: Prentice Hall; 2000. [2] Chawla TC, Chan SH. Combined radiation and convection in thermally developing Poiseuille flow with scattering. J Heat Transfer 1980;102:297–302. [3] Tabanfar S, Modest MF. Combined radiation and convection in absorbing, emitting, nongray gas-particulate tube flow. J Heat Transfer 1987;109:478–84. [4] Yucel A, Acharya S, Williams ML. Natural convection and radiation in a square enclosure. Numer Heat Transfer Part A 1989;15:261–78. [5] Lauriat G. Combined radiation–convection in gray fluids enclosed in vertical cavities. J Heat Transfer 1982; 104:609–15. [6] Chang LC, Yang KT, Lloyd JR. Radiation–natural convection interaction in two-dimensional complex enclosure. J Heat Transfer 1983;105:89–95. [7] Akiyama M, Chong QP. Numerical analysis of natural convection with surface radiation in a square enclosure. Numer Heat Transfer Part A 1997;32:419–33. [8] Riley N. Magnetohydrodynamics free convection. J Fluid Mech 1964;18:577–86. [9] Hunt JCR. Magnetohydrodynamic flow in a rectangular duct. J Fluid Mech 1965;21:577–90. [10] Buhler L. Laminar buoyant magnetohydrodynamic flow in a vertical rectangular ducts. Phys Fluids 1998; 10:223–36. [11] Shercliff JA. Steady motion of conducting fluids in pipes under transverse magnetic fields. Proc Camb Phil Soc 1953;49:136–44. [12] Alboussiere T, Garandet JP, Moreau R. Buoyancy-driven convection with a uniform magnetic field. Part 1, Asymptotic analysis. J Fluid Mech 1983;253:545–63. [13] Ghaly AY. Radiation effects on a certain MHD free-convection flow. Chaos Solitons Fractals 2002;13:1843–50. [14] Chamkha AJ, Issa C, Khanfer K. Natural convection from an inclined plate embedded in a variable porosity porous medium due to solar radiation. Int J Thermal Sci 2002;41:73–81. [15] Raptis A, Kafoussias N. Heat transfer in flow through a porous medium bounded by an infinite vertical plane under the action of magnetic field. Energy Res 1982;6:241–5. [16] Raptis A. Flow through porous medium in the presence of a magnetic field. Energy Res 1986;10:97–107. [17] Garandet JP, Alboussiere T, Moreau R. Buoyancy driven convection in a rectangular enclosure with a transverse magnetic field. Int J Heat Mass Transfer 1992;35:741–51. [18] Chamkha AJ. Unsteady hydromagnetic natural convection in a fluid-saturated porous medium channel. Adv Filtr Sep Technol 1996;10:369–79. [19] Bejan A. Entropy generation minimization. New York: CRC Press; 1996. [20] Arpaci VS. Radiative entropy production—heat lost to entropy. Adv Heat Transfer 1991;21:239–76. [21] Nield DA, Bejan A. Convection in porous media. New York: Springer-Verlag; 1998. [22] Bejan A. Convection heat transfer. New York: John Wiley & Sons; 1984.
Mixed convection–radiation interaction in a vertical porous channel: Entropy generation Shohel Mahmud , Roydon Andrew Fraser Department of Mechanical Engineering, University of Waterloo, 200 University Avenue West, Waterloo, On., Canada N2L 3G1 Received 30 October 2002
Abstract The present work examines analytically the effects of radiation heat transfer on magnetohydrodynamic mixed convection through a vertical channel packed with fluid saturated porous substances. First and Second Laws of thermodynamics are applied to analyze the problem. Special attention is given to entropy generation characteristics and their dependency on the various dimensionless parameters, i.e., Hartmann number (Ha), Plank number (Pl), Richardson number (Ri), group parameter (Br/II), etc. A steady-laminar flow of an incompressible-viscous fluid is assumed flowing through the channel with negligible inertia effect. The fluid is further considered as an optically thin gas and electrically conducting. Governing equations in Cartesian coordinates are solved analytically after reasonable simplifications. Expressions for velocity, temperature, local, and average entropy generation rates are analytically derived and presented graphically. # 2003 Elsevier Ltd. All rights reserved.
1. Introduction Any substance with a temperature above absolute zero transfers heat in the form of radiation. Thermal radiation always exits and can strongly interact with convection in many situations of engineering interest. The influence of radiation on natural or mixed convection is generally stronger than that on forced convection because of the inherent coupling between temperature and flow field (see [1]). Convection in a channel (or enclosed space) in the presence of thermal radiation continues to receive considerable attention because of its importance in many practical
Corresponding author. Tel.: +1-519-888-4567x3885; fax: +1-519-888-6197. E-mail address: [email protected] (S. Mahmud).
0360-5442/$ - see front matter # 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0360-5442(03)00154-3
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Nomenclature A B0 Be Br Br CK CP Eb Ec g G Gr Ha I k K n NF NH NS P P Pl Pr qR Re 000 S_ 000 Sc T u U v V w x X y Y
inverse Darcy number ¼ ve2 w=ðKU0 Þ magnetic induction (Wb m2) Bejan Bejan number ¼ NH =NS Brinkman number ¼ Ec Pr pffiffiffiffi modified Brinkman number ¼ Brw= K Forchheimer coefficient specific heat (kJ kg1 K1) blackbody radiation ¼ rT 4 Eckert number ¼ U02 =ðCP DTÞ gravitational acceleration (m s2) 0 a parameter ¼ qP =@X þ U 3 Grashof number ¼ ðgbDTw Þ=m2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Hartmann number ¼ B0 w re =ðqmÞ intensity of radiation thermal conductivity (W m1 K1) permeability (m2) an index ¼ 0 or 1 fluid friction irreversibility irreversibility due to heat transfer 000 000 entropy generation number ¼ S_ =Sc dimensionless pressure ¼ p=ðqf U02 Þ modified pressure ¼ P Re Plank number ¼ 16rT03 =k Prandtl number ¼ m=a radiative heat flux Reynolds number ¼ U 0 w=m entropy generation rate (W m3 K1) characteristic entropy transfer rate ¼ kDT 2 =ðw2 T02 Þ temperature ( C) fluid velocity in x direction (m s1) dimensionless axial velocity ¼ u=U 0 fluid velocity in y direction (m s1) dimensionless v velocity ¼ v=U 0 channel width (m) axial distance (m) dimensionless axial distance ¼ x=w transverse distance (m) dimensionless transverse distance ¼ y=w
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Greek a b e eL ; eR m q qf r re U X H j s P W C
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symbols thermal diffusivity (m2 s1) thermal expansion coefficient (K1) porosity emissivity kinematic viscosity (m2 s1) reflectivity density of fluid (kg m3) Stefan–Boltzmann constant fluid electrical conductivity (X1 m1) irreversibility distribution ratio ¼ N F =NS solid angle (st. rad.) dimensionless fluid temperature monochromatic absorption coefficient (m1) dimensionless optical thickness ¼ jw dimensionless temperature difference ¼ DT=T 0 porous-magnet parameter (see Eq. (21)) pffiffiffiffi porous media inertia coefficient ¼ CK e2 w= K
Subscript and superscript 0 reference value L value at left wall R value at right wall av average value
applications such as furnaces, combustion chambers, cooling towers, rocket engines, and solar collectors. During the past several decades, a number of experiments and numerical computations have been presented for describing the phenomenon of natural (or mixed) convection in channels or enclosures. These studies aimed at clarifying the effect of mixed convection on flow and temperature regimes arising from variations in the shape of the channel (or enclosure), in fluid properties, in the transition to turbulence, etc. Chawla and Chan [2] studied the effect of radiation heat transfer on thermally developing Poiseuille flow. The interaction of thermal radiation with conduction and convection in thermally developing, absorbing–emitting, nongray gas flow in a circular tube is investigated by Tabanfar and Modest [3]. Natural convection–radiation interaction is studied by Yucel et al. [4] for a square cavity, Lauriat [5] for a vertical cavity, and Chang et al. [6] for a complex enclosure. Akiyama and Chong [7] numerically analyzed the influence of gray surface radiation on the convection of nonparticipating fluid in a rectangular enclosed space. A combined free and forced convection flow of an electrically conducting fluid in a channel, in the presence of a magnetic field, is also of special technical significance because of its frequent occurrence in many industrial applications such as cooling of nuclear reactors, MHD marine
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propulsion, electronic packages, micro electronic devices, etc. Some other quite promising applications are in the field of metallurgy such as MHD stirring of molten metal and magnetic-levitation casting. Riley [8] studied buoyancy induced flow and transport in the presence of a magnetic field. For a rectangular vertical duct, Hunt [9] and Buhler [10] analyzed the fluid flow problem in a magnetic field with or without buoyancy effects. For a conducting fluid, Shercliff [11] analyzed the fluid flow characteristics in a pipe under a transverse magnetic field. Alboussiere et al. [12] did an asymptotic analysis to study buoyancy driven convection in a uniform magnetic field. Recently, Ghaly [13] and Chamkha et al. [14] analyzed the effect of radiation heat transfer on flow and thermal fields in the presence of a magnetic field for horizontal and inclined plates. References [15–18] presented some important aspects of magnetohydrodynamic free and/or mixed convection in a porous channel or enclosure. The foregoing discussions that deal with free, forced, and mixed convection problems of different geometries with or without radiation effect, in the presence and absence of a magnetic field in a porous or nonporous channel (or enclosure) are very much restricted to a First Law (of thermodynamics) analysis. The contemporary trend in the field of heat transfer and thermal design is to apply a Second Law (of thermodynamics) analysis, and its design-related concept of entropy generation minimization (EGM) (see [19]). This new trend is important and, at the same time, necessary, if the heat transfer community is to increase its contribution to viable engineering solutions to energy problems. Entropy generation is associated with thermodynamic irreversibility, which is present in all heat transfer processes. Because of the abundance of publications on entropy, no attempt is made here to review the literature; see Bejan [19] for a review on the application to problems of convection heat transfer and Arpaci [20] for radiation heat transfer. In this paper, we mainly focused on the entropy generation characteristics for a mixed convection flow inside a vertical porous channel with radiation heat transfer and under the action of transverse magnetic field.
2. Problem formulation Consider an optically thin and electrically conducting fluid flowing through a vertical channel packed with porous substance as shown in Fig. 1. The depth of the channel (along z-axis) is assumed sufficiently long compared to other dimensions. Both walls are isothermal and kept at the same or different temperatures. IL!R and IR!L represent the radiative intensity of the left and right walls, respectively. A transverse magnetic field of intensity B0 acts along the positive y-axis. Modeling the flow as ‘‘Boussinesq-incompressible’’ to take into account the coupling between the energy and momentum equations, we regard the density qf as constant everywhere except in the buoyancy term of momentum equation (Eq. (2)). The magnetic Reynolds number is assumed to be small so that the induced magnetic field is negligible, while the Hall effect of magnetohydrodynamics is assumed to be negligible. The flow through the porous medium is assumed to follow the Brinkman–Forchheimer model (see [21]) with additional body forces (buoyancy and magnetic forces). Correspondingly, the equations governing the steady-state con-
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Fig. 1. Schematic diagram of the problem under consideration.
servation of mass, momentum, and energy in the channel of Fig. 1 are @u @v þ ¼ 0; @x @y
me CK e 1 @u @u @2u 1 @p erB20 u; u þ pffiffiffiffi ujuj þ u þv ¼ egbðT T0 Þ þ m 2 K e @x @y @y qf @x qf K 2 @T @T @ T @2T 1 @qR þv ¼a ; u þ @x @y @x2 @y2 qf CP @y
(1) (2) (3)
where e, K, CK, b, re, B0, m, and a represent porosity, permeability, Forchheimer coefficient, volumetric thermal expansion coefficient, fluid electrical conductivity, magnetic flux, fluid kinematic viscosity, and thermal diffusivity, respectively. qR represents the radiative heat flux. A simplified approximation for calculating the radiative heat transfer term (qqR/@y) is presented in the next section before giving an approximate analytical solution to Eqs. (1)–(3).
3. Approximation of qqR/qy Most fluids of technological importance may be adequately described by the thin gas assumption. The behavior of thin gases near the boundary (solid vertical walls) is the knowledge we need for the present problem. Because of its negligible absorption, a thin gas is usually influenced by the geometry of the enclosures, and its behavior near a boundary depends on this
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geometry as well as on the boundary itself. A simplified mathematical model can be proposed for approximating qqR/@y term, according to Arpaci et al. [1], as follows: ð @qR (4) ¼ 4jEb j I dX; @y X where j, X, and Eb are the monochromatic absorption coefficient, solid angle, and blackbody emissive power, respectively. The second term at the right hand side of Eq. (4) represents the effect of boundaries. Assuming hemispherical isotropy, and replacing actual intensity with twostream intensities (outward and inward) as shown in Fig. 1, the second term of the right hand side of Eq. (4) can be written as ð I dX 2pðIL!R þ IR!L Þ: (5) X
Consider I0,L, I0,R are the intensities of radiation, eL ; eR are the emissivities, and qL, qR reflectivities for the left and the right walls, respectively. For a thin gas the absorption is negligible, and IL!R and IR!L remain uniform across the thickness of the gas between the plates (Fig. 1). Consequently IL!R and IR!L become identical. Approximating each stream as a combination of emission ðeI0 Þ and reflection (qI), IL!R and IR!L can be written as IL!R ¼ eL I0L þ qL IR!L
(6a)
IR!L ¼ eR I0R þ qR IL!R
(6b)
Now solutions to Eqs. (6a,b) for IL!R and IR!L yield IL!R ¼
eL I0L þ qL eR I0R ; 1 qL qR
IR!L ¼
eR I0R þ qR eL I0L : 1 qL qR
Substituting these intensities into Eq. (5) and transmissivity ¼ 0) and pI ¼ Eb , we get ð ð1=eR 1=2ÞEbL þ ð1=eL 1=2ÞEbR : I dX 4 ð1=eR 1=2Þ þ ð1=eL 1=2Þ X
(7) noting
that
qþe¼1
(assuming
(8)
Assuming that the emissivities of the left and right walls are identical ðeL ¼ eR Þ, substitution of Eq. (8) into Eq. (4) yields @qR 1 ¼ 4j Eb ðEbL þ EbR Þ : (9) 2 @y In the above expression, ðEbL þ EbR Þ=2 can be replaced by a mean emissive power Eb0. The two terms, Eb and Eb0, can be further expressed as rT4 and rT04 , where r is the Stefan– Boltzmann constant. The big difficulty arises due to the presence of the ‘‘T4’’ term in the energy equation, which essentially introduces a high nonlinearity and makes it impossible to obtain any analytical solution. Expressing T as a Taylor series about T0 with the assumption of small
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jTL TR j yields 1 X ðT T0 Þn dn 4 T ¼ f ðTÞ n! dT n T¼T0 n¼0 ¼ T04 þ 4ðT T0 ÞT03 þ
12ðT T0 Þ2 2 24ðT T0 Þ3 24ðT T0 Þ4 T0 þ T0 þ : 2 6 24
ð10Þ
Truncating the above series after the second term and using the definitions of Eb and Eb0, Eqs. (9) and (10), gives @qR
16jrT03 ðT T0 Þ: @y
(11)
4. Analysis The governing equations are put into their dimensionless forms with appropriate scaling parameters before giving velocity and temperature solutions. Lengths (x and y) are scaled with the channel width w. Velocity components (u and v) are scaled with the inlet velocity U0. The scaling factor for pressure is qU02 . Dimensionless temperature, H, is defined as ðT T0 Þ=DT, where T0 and DT are the reference temperature and reference temperature difference, respectively. For reference temperature T0, we chose ðTR þ TL Þ=2. For the reference temperature difference, DT, assuming that TR > T0 , we chose ðTR T0 Þ. The dimensionless forms of Eqs. (1)–(3) are then @U @V þ ¼ 0; @X @Y 1 @U @U @P 1 @2U @2U Gr Ha2 2 U þV þ U; þ H þ þ AU þ CU ¼ e @X @Y @X Re @X 2 @Y 2 Re2 Re 2 @H @H 1 @ H @2H Pl s U þV ¼ H; þ @X @Y RePr @X 2 @Y 2 RePr
(12) (13) (14)
where A, C, Re, Pr, Gr, Ha, and Pl represent inverse Darcy number, porous media inertia coefficient, Reynolds, Prandtl, Grashof, Hartmann, and Plank numbers, respectively. s is the dimensionless optical thickness. The definitions of the above parameters are given in the nomenclature section. The nonlinear nature of Eqs. (13) and (14) restricts us to solving them for a close, not exact, form of an analytical solution. Assume now that the flow is hydrodynamically and thermally fully developed (qU=qX ¼ 0, qH=qX ¼ 0). When combined with the continuity equation (Eq. (12)) and the impermeable boundary condition at the channel surface, we obtain V ¼ 0. 0 ) yields Expansion of U2 as a Taylor series about the dimensionless inlet velocity (U 1 X 0 Þn dn ðU U 2 2 þ 2ðU U 0 ÞU 0 þ ðU U 0 Þ2 : U ¼ f ðUÞ ¼U (15) 0 n n! dU 0 U¼U n¼0 Truncating the above series after the second term gives a reasonable approximation for U2 term in the momentum equation (last term on the left-hand side of Eq. (13)). After simplifi-
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cation, Eqs. (13) and (14) become pffiffiffiffiffiffiffiffiffiffiffi @2U @P 2 0 þ Ha2 U; GrRiH þ ARe þ 2CReU ¼ þ U 0 2 @Y @X @2H Pl sH ¼ 0; @Y 2
(16) (17)
where modified pressure ðP Þ is the product of dimensionless pressure (P) and Reynolds number (Re). Boundary conditions are At Y ¼ 0;
U ¼ 0 and H ¼ HL ; At Y ¼ 1;
U ¼ 0 and H ¼ HR :
(18)
Expressing the square root of the product of Plank number (Pl) and optical thickness (s) as the radiation parameter (Rd), the solution to Eq. (17) for the temperature H, subject to the thermal boundary conditions defined in Eq. (18), yields H ¼ C1 sinhðRd Y Þ þ C2 coshðRd Y Þ;
(19)
where C1 and C2 are two constants of integration and can be expressed by the following equation: 8 HR HL coshðRd Þ < ; C1 ¼ (20) sinhðRd Þ : C 2 ¼ HL : With the solution for H already determined, Eq. (16) can be solved for the velocity U subject to the no-slip boundary conditions given in Eq. (18). Omitting the detail arithmetic operations, the expression for the dimensionless velocity U becomes pffiffiffiffiffiffiffiffiffiffiffi C1 sinhðRd Y Þ þ C2 coshðRd Y Þ G (21) þ C3 sinhðWY Þ þ C4 coshðWY Þ 2 ; U ¼ GrRi 2 2 W W Rd where pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 þ Ha2 ; W ¼ ARe þ 2CReU
@P 2 G¼ þ U0 ; @X
where Richardson number (Ri) is the ratio between Grashof number (Gr) and square of the Reynolds number (Re2). Richardson number is an important parameter for characterizing mixed convective flows. It determines the relative dominance between buoyancy, viscous, and inertia forces. The constants C3 and C4 of Eq. (21) can be expressed as 1 G C1 sinhðRd Þ þ C2 coshðRd Þ C3 ¼ C coshðWÞ ; 4 sinhðWÞ W2 W2 R2d (22) G C2 : C4 ¼ 2 2 W W R2d In the above equation W is termed as a porous-magnet parameter. It is interesting to note that for equal magnitudes of W and Radiation parameter (Rd), the constants C3 and C4 are singular (1). This is an important restriction (W 6¼ Rd ) for the present problem. The magnitude
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of Reynolds number in mixed convection problems is usually kept low. For a porous medium with comparatively low porosity ðeÞ and high permeability (K), the parameter W reduces to the Hartmann number (see Eq. (21)). The dimensionless form of the average velocity along the X direction, Uav, can be evaluated by integrating Eq. (21) with respect to Y (0 Y 1), and yields pffiffiffiffiffiffiffiffiffiffiffi GrRi ½C CoshðWÞ þ C4 SinhðWÞ G
½C1 CoshðRd Þ þ C2 SinhðRd Þ þ 3 Uav ¼ 2 2 2 W W Rd W Rd pffiffiffiffiffiffiffiffiffiffiffi C3 C1 GrRi
: (23) W Rd W2 R2d
5. Second Law analysis The convection process in a channel is inherently irreversible. Nonequilibrium conditions arise due to the exchange of energy and momentum, within the fluid and at the solid boundaries. This causes continuous entropy generation. One part of this entropy production is due to the heat transfer in the direction of finite temperature gradients, which is common in all types of thermal engineering applications (see [19]). Another part of the entropy production arises due to the fluid friction and is generally termed as fluid friction irreversibility. Based on the assumptions already made, the volumetric rate of entropy generation for the present problem can be expressed as 2 2 _S000 ¼ k @T þn l @u þð1 nÞ l u2 : (24) 2 T0 @y KT0 T0 @y For a non-Darcy model of the porous media, the first two terms in Eq. (24) are used to calculate the local entropy generation rate (n ¼ 1). For a channel with a porous media that obeys Darcy’s model, entropy generation due to fluid friction can be evaluated using the third term (n ¼ 0) on the right hand side of Eq. (24). For details, see [19,22]. Entropy generation rate is positive and finite as long as temperature and velocity gradients are present in the medium. According to Bejan [19], the dimensionless form of the entropy generation rate is the entropy generation number (NS) which, by definition, equals the ratio of 000 actual entropy generation rate (S_ ) to the characteristic entropy transfer rate ðkDT 2 =ðw2 T02 ÞÞ. Using the same parameters, already used for scaling purpose, the dimensionless form of Eq. (24) is @H 2 Br @U 2 Br NS ¼ þn þð1 nÞ (25) ½U 2 : @Y P @Y P In Eq. (25), Br and P are Brinkman number and dimensionless temperature difference pffiffiffiffi (DT/ T0), respectively. The modified Brinkman number ðBr Þ is the product of Br and w= K . The ratio Br/P is commonly termed the group parameter. Combining Eqs. (19), (21) and (25), the
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dimensionless form of entropy generation number (NS) can be written as Br NS ¼ R2d ½C1 coshðRd Y Þ þ C2 sinhðRd Y Þ2 þn P " pffiffiffiffiffiffiffiffiffiffiffi #2 GrRi fC1 coshðRd Y Þ þ C2 sinhðRd Y Þg þ WfC3 coshðWY Þ þ C4 sinhðWY Þg W2 R2d þ ð1 nÞ
Br 2 U : P
ð26Þ
The first term at the right hand side of Eq. (26) represents the heat transfer part of entropy generation (NH) and the second part is the fluid friction contribution to entropy generation (NF).
6. Results and discussion It is difficult to study the influence of all parameters involved in the present problem on the flow and thermal field along with entropy generation characteristics. Therefore, a selected set of graphical results are presented in Figs. 2–14 that will provide a good understanding of the influence of different parameters on the velocity, temperature, and entropy generation profiles. Both symmetrical ðHL ¼ 1; HR ¼ 1Þ and asymmetrical temperatures ðHL ¼ 0; HR ¼ 1Þ at the walls are considered.
Fig. 2. Variation of velocity profile with radiation parameter.
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Fig. 3. Variation of velocity profile with porous-magnet parameter.
Fig. 4. Variation of temperature profile with radiation parameter.
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Fig. 5. Variation of entropy generation number with group parameter.
Fig. 6. Variation of entropy generation number with radiation parameter.
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Fig. 7. Variation of entropy generation number with porous-magnet parameter.
Fig. 8. Variation of entropy generation number with mixed convection parameter.
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Fig. 9. Average entropy generation as a function of radiation parameter.
Fig. 10. Variation of average entropy generation with mixed convection parameter at n ¼ 1.
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Fig. 11. Variation of average entropy generation with mixed convection parameter at n ¼ 0.
Fig. 12. Minimum entropy generation at different group parameters for n ¼ 1.
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Fig. 13. Minimum entropy generation at different group parameters for n ¼ 0.
Fig. 14. Rd,opt–(Gr Ri)0.5 relation at different group parameters for n ¼ 1.
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The roles of inverse Darcy number (A), porous media inertia coefficient (C), and Hartmann number (Ha) are very similar for the present problem. These parameters essentially add extra resistance to the flow field. Instead of analyzing individual influences, we analyze their influence as a group (influence of W). It should be noted that for a limiting case (low porosity and high permeability), the parameter W reduces to the Hartmann number (Ha). 6.1. Velocity distribution Modified dimensionless velocity ðU Þ is calculated by dividing the velocity U by the average velocity Uav. Fig. 2 shows the variation of U as a function of Y at different radiation parameters (Rd) associated with the case of constant and symmetrical wall temperatures. Velocity profiles are symmetrical about the centerline (Y ¼ 0:5) of the channel for each value of Rd. Increases in the value of Rd have a tendency to slow down the movement of the fluid in the middle portion of the channel. For Rd ¼ 6, the velocity profile is almost flat. For Rd ¼ 7 and 8, a maximum velocity occurs at two different locations (not at Y ¼ 0:5). Velocity gradient (qU/ qY) becomes zero at three different locations for Rd ¼ 7 and 8, which have zero contribution to entropy generation due to fluid flow (will discuss later in detail). The effect of the porous-magnet parameter (W) on the velocity profile is shown in Fig. 3 for symmetric temperatures ðHL ¼ HR ¼ 1Þ at the walls. The parameter W creates a resistive force similar to a drag force; it acts in the opposite direction of the fluid motion, thus causing the velocity of the fluid to decrease. This flow resistance is depicted by the decreases of the U as W increases in Fig. 3. 6.2. Temperature distribution Dimensionless temperatures (H) are plotted as a function of transverse distance Y in Fig. 4 for six selected values of radiation parameters. Temperatures at the walls are kept the same ðHL ¼ HR ¼ 1Þ. For Rd ¼ 0, Eq. (17) becomes a simple conduction equation and the corresponding temperature profile is similar to a conduction temperature profile. Temperature does not vary along the transverse distance and H ¼ 1 everywhere. For Rd > 0, the profiles become nonlinear but symmetric about the centerline of the channel. Hmin occurs at Y ¼ 0:5 and the magnitude of Hmin decreases with increasing Rd. For the symmetric temperature boundary condition ðHL ¼ HR Þ, the temperature gradient (qH/qY) is zero at the channel’s centerline where no heat transfer entropy is generated since there is no finite temperature gradient. 6.3. Idle point for entropy generation The location where the temperature gradient (qH/qY) is zero is an idle point for temperature entropy generation. No entropy generation occurs at points of zero temperature gradient. The location of such a point can be mathematically expressed by the following equation: 1 HR HL coshðRd Þ HL sinhðRd Þ ln : (27) Y j@H=@Y ¼0 ¼ 2Rd HR HL coshðRd Þ þ HL sinhðRd Þ
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It should be noted that, for a particular combination of HL, HR, and Rd, an idle point for temperature entropy may or may not exist. Any solution to Eq. (27) which yields an imaginary number, a value less than 0, or a value greater than 1, leads to a nonexisting idle point for temperature entropy. Similarly fluid friction irreversibility becomes zero at the location of zero velocity gradient (qU/qY). The complicated expression of velocity (see Eq. (21)) makes it difficult to derive an analytical expression (like Eq. (27)) for a fluid friction irreversibility idle point. An alternative approximate solution can be achieved by expanding the velocity gradient (qU/qY) in a series like @U X gC1 DRnd þ g C2 DRnd þ gC3 Wnþ1 þ g C4 Wnþ1 n ¼ n ¼ 0; 1; 2; . . . (28) Y ; @Y n! n where pffiffiffiffiffiffiffiffiffiffiffi Rd GrRi ; D¼ 2 W R2d
g ¼ 1 g;
1; when n is 0 or odd number; g ¼ 0; when n is even number
truncating the series after a desired order of n, set this truncated series to zero, and solving for Y ð¼ Y jqU=@Y ¼0 Þ. The accuracy of the solutions depends on n. 6.4. Entropy generation rate The foregoing discussions, which mainly focused on the flow field and thermal field characteristics in terms of velocity and temperature distribution profiles, are important because of their (velocity and temperature) close relation to entropy generation rate (see Eqs. (24) and (25)). Next we turn our attention to local and global entropy generation rates and EGM. Four parameters (Br/P, Rd, W, and (Gr Ri)0.5) are selected to show their influence on the local rate of entropy generation through graphical representation in Figs. 5–8. For each case, symmetric wall temperatures ðHL ¼ HR ¼ 1Þ are selected for the thermal boundary condition. Group parameter (Br/P) is an important dimensionless number for entropy generation analysis. It determines the relative importance of viscous effect to temperature entropy generation. Entropy generation number (NS) is plotted as a function of Y at different group parameters in Fig. 5. The entropy generation profile is symmetric about the centerline of the channel due to the symmetric distributions of velocity and temperature. For all group parameters, each wall acts as a strong concentrator of entropy generation because of the high near-wall gradients of velocity and temperature. For all Br/P, no entropy is generated at the centerline of the channel. Both velocity and temperature gradients (qU/@Y and qT/@Y) are zero at the centerline leaving no contribution to entropy generation. Due to high viscous effects, higher Br/P shows a larger NS at a particular location of Y. Fig. 6 presents the effect of radiation parameter (Rd) on local entropy generation rate. Entropy generation is characterized by concave shaped and symmetric profiles for all values of Rd. Small changes in Rd cause large variations in NS as shown in Fig. 6. Porous-magnet parameter (W) does not dominate like the radiation parameter. For small values of W, some variation is observed in the entropy generation profiles due to changes in W as shown in Fig. 7. This variation is almost negligible at higher values of W. Finally, in Fig. 8, the effect of mixed convection
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parameter ((Gr Ri)0.5) on entropy generation number is tested. A small range (0.1–5.0) of (Gr Ri)0.5 is selected. Entropy generation distribution is still characterized by a concave shaped profile. A small change in the value of (Gr Ri)0.5 causes a rapid rise in entropy generation. At a particular location of Y, NS increases with the increases in (Gr Ri)0.5. 6.5. Average entropy generation From the expression for entropy generation number (Eq. (26)), the volume averaged entropy generation rate can be evaluated using the following equation: ð 1 NS d8 ¼ NH;av þ NF;av ; (29) NS;av ¼ 8 8 where 8 is the volume of the channel. Average entropy generation number (NS,av) consists of two parts: the first part, NH,av, is the contribution of heat transfer, and the second part, NF,av, is the contribution of fluid friction. For a particular case, Fig. 9 shows the variation of average entropy generation rate with radiation parameter. Contributions to the total entropy generation are always an internal competition between irreversibilities due to heat transfer and fluid friction. For the selected range of Rd, NF,av decreases and NH,av increases with the increasing Rd. This opposite behavior of heat transfer and fluid friction irreversibilities with change in Rd leaves an optimum radiation parameter (Rd,opt) where total entropy generation in minimum. Fig. 10 shows the variation of NS,av with radiation parameter (Rd) at different mixed convection parameters (Gr Ri)0.5 and n ¼ 1. The reason behind choosing of parameters (Gr Ri)0.5 and Rd is due to their practical importance and strong contribution to entropy generation. For a particular (Gr Ri)0.5, NS,av decreases with increasing of Rd, and shows its minimum ([NS]min) at a particular value of Rd, then increases. For the selected range of (Gr Ri)0.5, NS,av profiles merge with each other after Rd 15. For n ¼ 0, Fig. 11 shows the distribution of NS,av with Rd while keeping other parameters the same as in Fig. 10. Profiles are similar to those in Fig. 10 except that the variation of magnitude of NS,av differs. 6.6. Entropy generation minimization The contemporary trend in the field of heat transfer and thermal design is Second Law (of thermodynamics) analysis and its design-related concept of entropy generation and its minimization ([19]). EGM is the method of modeling and optimization of real devices that owe their thermodynamic imperfection to heat transfer, mass transfer, and fluid flow irreversibilities. It is also known as thermodynamic optimization in engineering, where it was first developed, or more recently as finite space–time thermodynamics in the physics literature. The method combines from the start the most basic principles of thermodynamics, heat transfer, and fluid mechanics. Entropy generation minimization combines into simple models the most basic concepts of heat transfer, fluid mechanics, and thermodynamics. These simple models are used in the optimization of real (irreversible) devices and processes, subject to finite-size and finite-time constraints that are in fact responsible for the irreversible operation of the device. The combined heat transfer and thermodynamics model ‘‘visualizes’’ for the analyst the irreversible nature of the device. So, for proper use of this method, an analyst must know the behavior of system’s
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Fig. 15. Rd,opt–(Gr Ri)0.5 relation at different group parameters for n ¼ 0.
irreversibilities in terms of minimum entropy generation and the variation of irreversibilities with system parameters (flow parameters, transport properties, geometry, etc.). Figs. 12–15 present the important aspect of ‘‘entropy generation minimization (EGM)’’. The EGM criterion is the most appropriate measure of a real system’s approach to thermodynamic ideality (see [19]) and at the same time is a method of thermodynamic optimization for real systems. For n ¼ 1, [NS]min is plotted as a function of (Gr Ri)0.5 in Fig. 12 at different group parameters (Br/P). For all Br/P, [NS]min gradually increases with (Gr Ri)0.5. Fig. 13 shows the same distribution for n ¼ 0. The corresponding magnitude of the radiation parameter at which entropy becomes a minimum is termed Rd,opt, and which is plotted as a function of (Gr Ri)0.5 in Fig. 14 for n ¼ 1 and in Fig. 15 for n ¼ 0.
7. Conclusions We investigated analytically the First and Second Laws (of thermodynamics) aspects of fluid flow and heat transfer inside a vertical porous channel with a transverse magnetic field. Radiation heat transfer is considered assuming the fluid to be an optically thin gas. The influence of different dimensionless parameters, i.e. Rd, W, (Gr Ri)0.5, and Br/P, is tested on the calculated velocity (U), temperature (H), and local and average entropy generation number. Radiation parameter (Rd) has a strong influence on velocity and temperature. Higher values of Rd and W suppress the velocity profile around the centerline of the channel. The radiation parameter introduces nonlinearity in temperature profiles. An increase in the value of Rd shifts (in a non-
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linear fashion) the temperature profile away from the conduction profile (the linear one) which occurs at Rd ¼ 0. Entropy generation number is characterized by a concave shaped profile and is symmetric about the channel centerline for a symmetrical temperature boundary condition. Radiation and mixed convection parameters have a more dominating influence on entropy generation rate than porous-magnet and group parameters. Expressions for the idle points of entropy generation are derived and their location(s) are determined. Irreversibility due to the heat transfer and/or fluid friction become zero at these idle points. Based on entropy generation minimization, optimum radiation parameters (Rd,opt) are determined which increase with increasing (Gr Ri)0.5 and Br/P ðor Br =PÞ).
References [1] Arpaci VS, Selamet A, Kao SH. Introduction to heat transfer. New York: Prentice Hall; 2000. [2] Chawla TC, Chan SH. Combined radiation and convection in thermally developing Poiseuille flow with scattering. J Heat Transfer 1980;102:297–302. [3] Tabanfar S, Modest MF. Combined radiation and convection in absorbing, emitting, nongray gas-particulate tube flow. J Heat Transfer 1987;109:478–84. [4] Yucel A, Acharya S, Williams ML. Natural convection and radiation in a square enclosure. Numer Heat Transfer Part A 1989;15:261–78. [5] Lauriat G. Combined radiation–convection in gray fluids enclosed in vertical cavities. J Heat Transfer 1982; 104:609–15. [6] Chang LC, Yang KT, Lloyd JR. Radiation–natural convection interaction in two-dimensional complex enclosure. J Heat Transfer 1983;105:89–95. [7] Akiyama M, Chong QP. Numerical analysis of natural convection with surface radiation in a square enclosure. Numer Heat Transfer Part A 1997;32:419–33. [8] Riley N. Magnetohydrodynamics free convection. J Fluid Mech 1964;18:577–86. [9] Hunt JCR. Magnetohydrodynamic flow in a rectangular duct. J Fluid Mech 1965;21:577–90. [10] Buhler L. Laminar buoyant magnetohydrodynamic flow in a vertical rectangular ducts. Phys Fluids 1998; 10:223–36. [11] Shercliff JA. Steady motion of conducting fluids in pipes under transverse magnetic fields. Proc Camb Phil Soc 1953;49:136–44. [12] Alboussiere T, Garandet JP, Moreau R. Buoyancy-driven convection with a uniform magnetic field. Part 1, Asymptotic analysis. J Fluid Mech 1983;253:545–63. [13] Ghaly AY. Radiation effects on a certain MHD free-convection flow. Chaos Solitons Fractals 2002;13:1843–50. [14] Chamkha AJ, Issa C, Khanfer K. Natural convection from an inclined plate embedded in a variable porosity porous medium due to solar radiation. Int J Thermal Sci 2002;41:73–81. [15] Raptis A, Kafoussias N. Heat transfer in flow through a porous medium bounded by an infinite vertical plane under the action of magnetic field. Energy Res 1982;6:241–5. [16] Raptis A. Flow through porous medium in the presence of a magnetic field. Energy Res 1986;10:97–107. [17] Garandet JP, Alboussiere T, Moreau R. Buoyancy driven convection in a rectangular enclosure with a transverse magnetic field. Int J Heat Mass Transfer 1992;35:741–51. [18] Chamkha AJ. Unsteady hydromagnetic natural convection in a fluid-saturated porous medium channel. Adv Filtr Sep Technol 1996;10:369–79. [19] Bejan A. Entropy generation minimization. New York: CRC Press; 1996. [20] Arpaci VS. Radiative entropy production—heat lost to entropy. Adv Heat Transfer 1991;21:239–76. [21] Nield DA, Bejan A. Convection in porous media. New York: Springer-Verlag; 1998. [22] Bejan A. Convection heat transfer. New York: John Wiley & Sons; 1984.