Energy 50 (2013) 143e149
Contents lists available at SciVerse ScienceDirect
Energy journal homepage: www.elsevier.com/locate/energy
Optimum slip flow based on the minimization of entropy generation in parallel plate microchannels Guillermo Ibáñez*, Aracely López, Joel Pantoja, Joel Moreira, Juan A. Reyes Universidad de Ciencias y Artes de Chiapas, Libramiento Norte Poniente No. 1150, Col. Lajas Maciel, Tuxtla Gutiérrez, Chiapas, C.P. 29000, Mexico
a r t i c l e i n f o
a b s t r a c t
Article history: Received 18 June 2012 Received in revised form 27 November 2012 Accepted 28 November 2012 Available online 4 January 2013
The effects of slip flow on heat transfer and entropy generation by considering the conjugate heat transfer problem in microchannels are studied, analytically. The heat transfer equations in the fluid and the finite thickness walls of the microchannel are solved analytically using thermal boundary conditions of the third kind at the outer surfaces of the walls and continuity of temperature and heat flux across the fluidewall interfaces. After the analytic solutions for the velocity and temperature fields in the fluid and walls of microchannel are obtained, the entropy generation rate is discussed in detail and investigated considering slip flow and convective effects, simultaneously. The results show that the global entropy generation rate is minimized for certain suitable combination of the geometrical and physical parameters of the system. It is possible to find an optimum slip velocity which leads to a minimum global entropy generation rate. The Nusselt number is also calculated and explored for different conditions. An optimum value of the slip length that maximizes the heat transfer is derived. Ó 2012 Elsevier Ltd. All rights reserved.
Keywords: Slip flow Optimization Entropy generation Irreversibility
1. Introduction The performance of thermal devices is always affected by irreversible losses that lead to an increase of entropy and reduce the thermal efficiency. Therefore, in the energy optimization problems and design of many traditional heat removal engineering devices, it is necessary to evaluate the entropy generation or exergy destruction due to heat transfer and viscous friction as a function of the design variables selected for the optimization analysis. A variety of fluid flow systems have been analyzed and optimized using the entropy generation minimization (EGM) method. This method, originally formulated by Bejan [1,2], has already been applied to heat exchangers [3], two-phase flows [4], fuel cells [5e7], magnetohydrodynamic (MHD) flows subjected to ohmic dissipation [8e 12], heat transfer problems with temperature dependent heat sources [13], cases where a non-uniform thermal conductivity exists [14], flows with temperature dependent fluid viscosity [15,16], natural convection flows [17e19], as well as nanofluids between co-rotating cylinders with constant heat flux on the walls [20]. A review of the literature shows that another optimization
* Corresponding author. Tel.: þ52 (0)1 961 6170440x4200; fax: þ52 (0)1 961 6170440x4231. E-mail addresses:
[email protected],
[email protected] (G. Ibáñez),
[email protected] (A. López),
[email protected] (J. Pantoja), jmoreira23@ yahoo.com.mx (J. Moreira). 0360-5442/$ e see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.energy.2012.11.036
procedure using multiobjective optimization techniques has been investigated to evaluate not only the entropy generation but also other functions, simultaneously. In particular, Ahmadi et al. [21] used a multi-objective genetic algorithm to minimize the entropy generation and the total annual cost in a cross flow plate fin heat exchanger, simultaneously. More recently, Guo et al. [22] introduced a new optimization theory as an alternative optimization criterion in heat transfer processes, termed as the entransy theory. Thereafter, based on this theory several recent works have been reported in the literature [23e26]. The results indicated that when the objective of a heat exchanger is for heating or cooling, the irreversibility should be measured by the entransy dissipation, while for heat-work conversion transfer processes, the irreversibility should be described by the entropy generation. For instance, Cheng et al. [25] compared the minimum entropy generation principle and the entransy dissipation extremum principle, and analyzed their physical implications and applicability in heat convection optimization. Their analysis indicated that the entransy dissipation extremum principle is more suitable to maximize the heat transfer performance, while the minimum entropy generation principle is more suitable for maximizing the heat-work conversion. In the present work we apply the EGM method to minimizing the entropy generation in parallel plate microchannels. In the recent years, the microfluidics became an attractive area of research. Analysis of heat and fluid flow at microscale is of great importance for its application in micro heat exchanger systems. Particularly, the study of steady laminar forced convection fluid flow
144
G. Ibáñez et al. / Energy 50 (2013) 143e149
through parallel plates in a microchannel has many significant applications in engineering. Starting from the design of cooling system for electronic devices to various microscale electromechanical systems, such as micro pumps and micro turbines, this type of geometry can be observed. Reducing entropy generation of fluid motion through microchannels can have a beneficial impact on the input power required to achieve both desired heat exchange and mass flow rates. Previous experimental and theoretical studies have identified the importance of entropy production in optimal designs of microfluidic devices. In particular, some recent works have addressed the analysis of entropy generation in microchannels considering mainly the effects of viscous and thermal irreversibilities [27e33]. In turn, the dissipative processes that arise in a microchannel flow subjected to electromagnetic interactions have been analyzed in Refs. [34e39]. Although the entropy generation rate is calculated in these microfluidic devices, “no-slip” boundary condition is considered for the velocity field. In some application in microfluidic and nanofluidic devices where the surface to volume ratio is large, the slip flow conditions are more typical and slip boundary condition is usually used for the velocity field [40e44]. The degree of slip is normally quantified through the slip length, defined as the distance from the liquid to the surface within the solid phase, where the extrapolated flow velocity vanishes [42]. In liquid flows, the slip length can be affected by many physicoechemical parameters, including wetting, surface roughness, impurities, shear rate, pressure, or temperature, among many others. Entropy generation analysis at microscale with the inclusion of slip velocity is reported by various authors. Ogedengbe et al. [45] have investigated the mechanisms of near-wall velocity slip and their effects on energy conversion of fluid motion in microchannels. They found that various design parameters (such as operating pressure and channel aspect ratios) can be modified to reduce entropy generation in microchannels for gas flows. In another task [46], they have performed a numerical study of slip flow irreversibility effects in a counterflow heated microchannel. The effects of channel size perturbation, Reynolds number, and pressure ratios on the exergy destruction are presented. Hooman [47] has studied the entropy generation for forced convection in microelectromechanical system in the slip flow regime. He analyzed two different cross sections: i) microducts (composed of two parallel plates) and ii) micropipes. It was found that dimensionless entropy generation number (Ns) always decreases with Knudsen number (Kn). Yazdi et al. [48] have analyzed the entropy generation in external liquid flow over a surface containing embedded parallel microchannels. The results showed that the rate of entropy generation always decreases with increasing slip length. In the present contribution, the main objective is to explore the effects of the slip flow condition on both the heat transfer and entropy generation, and to show the existence of optimal values of this quantity consistent with minimum entropy generation rates and maximum heat transfer in the thermal behavior of a viscous flow in a parallel wall microchannel. The conjugated heat transfer problem in the fluid and the finite thickness walls of the microchannel is solved analytically, and convective heat transfer is considered assuming a thermally fully developed flow. Exact analytic solutions are presented for the velocity and temperature
fields in the fluid and solid walls. It is noted that the present solution reduces to our previous results [49] by taking the slip parameter equal to zero which provides a useful mathematical check. In the present work the local Nusselt number is explored. An optimum value of the slip length that maximizes the heat transfer is derived. The above shows that the fundamental contributions of the work are: a) The entropy production is determined from the solution of conjugate heat transfer problem considering slip flow and convective heat transfer effects, simultaneously, b) All relevant design parameters for the system are optimized by minimizing the global entropy generation. Particularly, an optimum value of slip length that minimize the entropy generation is found, c) An optimum value of slip length that maximizes the heat transfer is obtained and d) The effects of slip flow on the optimum values of some other parameters are analyzed. In the following sections, the problem is formulated, analyzed, solved and discussed. Section 2 consists of the transport problem analysis which contains the momentum and energy balance equations and their solutions. Section 3 contains the determination of the entropy generation rate and the Nusselt number for this problem, and graphical representation of results and their discussion. Section 4 contains the concluding remarks. 2. Transport problem 2.1. Velocity field We consider the steady fully developed flow of a viscous fluid in a parallel wall microchannel of finite thickness in the presence of a constant longitudinal pressure gradient dp=dx0 . The upper wall is located at y0 ¼ a and the lower wall is at y0 ¼ a, y0 , denoting the transversal coordinate. We also assume that physical properties of the fluid and walls are constant and that the fluid is incompressible and monocomponent, so that the mass diffusion phenomena are disregarded. At the fluidewall interfaces, the slip condition on the velocity is applied. The momentum equation is
0 ¼
dp d2 u0 þ h 02 ; 0 dx dy
Let us assume that the surface roughness of each wall is in general different. Then, the slip lengths, although taken to be constant, do not have the same value on both walls. Therefore, Eq. (1) must satisfy the boundary conditions
u0 þ a01
du0 ¼ 0; dy0
at y0 ¼ a
(2)
u0 a02
du0 ¼ 0; dy0
at y0 ¼ a
(3)
Here, a and h are the separation between the walls and dynamic viscosity of the fluid, respectively, while a01 and a02 are the slip lengths of the upper and lower walls, respectively, which in general, are assumed to be different. The analytic solution of Eq. (1) that satisfies slip boundary conditions, Eqs. (2) and (3), is
2 3 4 0 0 0 2 0 þ a0 0 a a a 3 þ þ 2a 0 0 1 2 a a 2 1 2 1 2 dp 6 y y 7 a u0 ¼ þ 4 5 2a þ a01 þ a02 2n dx0 a 2a þ a01 þ a02 a a2
(1)
(4)
G. Ibáñez et al. / Energy 50 (2013) 143e149
The average velocity in the cross-section of the microchannel is given by
1 U ¼ 2a
Zþa a
2 3 4 0 0 0 0 a1 a2 þ 2a 17 2 dp 3 a1 þ a2 þ a 6 a u0 dy0 ¼ 5 4 3 2a þ a01 þ a02 2n dx0
145
The boundary conditions that Eqs. (6)e(8) have to satisfy are the following:
Q ¼ Qw1 ;
at y ¼ 1;
Q ¼ Qw2 ;
at y ¼ 1;
(9) (10)
Hence, in terms of the dimensionless variables u ¼ u0 =U, y ¼ y0 =a, a1 ¼ a01 =a and a2 ¼ a02 =a, Eq. (4) becomes
dQ dQ ¼ g1 w1 ; dy dy
at y ¼ 1;
(11)
i h 3 u ¼ F1 þ F2 y þ y2 3F1 1
dQ dQ ¼ g2 w2 ; dy dy
at y ¼ 1;
(12)
(5)
where
3ða1 þ a2 Þ þ 4ða1 a2 Þ þ 2 F1 ¼ ; 2 þ a1 þ a2 2ða1 a2 Þ F2 ¼ ; 2 þ a1 þ a2 With a1 ¼ a2 ¼ 0, the velocity profile given by Eq. (5) reduces to the known (no slip) Poiseuille flow through a parallel plate microchannel. 2.2. Temperature field Once the velocity field is known, the temperature fields for the fluid and solid walls are determined from the solution of the conjugate heat transfer problem. We proceed to solve the energy balance equation within the microchannel considering viscous dissipation and convective effects. In dimensionless terms, the heat transfer equation in the fluid reduces to [49]
ðPeÞu
2 vQ v2 Q du þ ; ¼ dy vx vy2
(6)
where the dimensionless temperature is now given by Q ¼ kðT Ta Þ=hU 2 , with T being the fluid temperature and k the fluid thermal conductivity. Here, Pe ¼ UarC=k is the Péclet number where r and C are the fluid density and the specific heat of the fluid, respectively. As in [38,49], the heat transfer equation in both walls of the microchannel is also required in the solution of conjugate heat transfer problem. For the lower and upper walls, respectively, we have
d2 Qw1 ¼ 0; dy2
C1 ¼
(7)
dQw1 Bi1 ðQw1 Qa Þ ¼ 0; dy
at y ¼ ð1 þ d1 Þ;
(13)
dQw2 þ Bi2 ðQw2 Qa Þ ¼ 0; dy
at y ¼ ð1 þ d2 Þ;
(14)
where Qa ¼ kTa =hU 2 is the dimensionless ambient temperature. Eqs. (9)e(12) express continuity conditions for the temperature and heat flux across the fluidewall interfaces, where g ¼ kw =k is the wall to fluid thermal conductivity ratio, kw being the thermal conductivity of the wall. In turn, conditions (13) and (14) at the outer surfaces of the walls establish that the amount of heat entering or leaving the system depends on the external temperature as well as on the convective heat transfer coefficient, expressed in dimensionless terms by the Biot number, Bii , for each wall ði ¼ 1; 2Þ [38]. In order to solve Eqs. (6)e(8) under boundary conditions (9)e (14), we only analyze the thermally fully developed region so that the dimensionless temperature can be expressed as Qðx; yÞ ¼ Ax þ qðyÞ, where the axial temperature gradient A is assumed to be constant [50]. The temperature fields for the fluid and solid walls are determined from the solution of the previous boundary value problem. For the fluid region, we get
qðyÞ ¼
3 2
4ð3F1 1Þ
h 6 F1 F3 F22 y2 þ 2F2 ð4 þ F3 Þy3 i ð4 F3 Þy4 þ C1 y þ C2 ;
In turn, the temperature in the microchannel walls is given by
Qw1 ¼ C3 y þ C4 ;
(16)
Qw2 ¼ C5 y þ C6 ;
(17)
where
F5 Bi1 g1 ðBi2 d2 þ 1Þ F4 Bi2 g2 ðBi1 d1 þ 1Þ 2F2 ð4 þ F3 ÞðBi1 Bi2 g1 g2 Þ ; 2Bi1 Bi2 g1 g2 þ Bi2 g2 ðBi1 d1 þ 1Þ þ Bi1 g1 ðBi2 d2 þ 1Þ 2ð3F1 1Þ2 3
d2 Qw2 ¼ 0; dy2
(8)
where Qwi ¼ kTwi =hU 2 is the dimensionless wall temperature. Subindexes i ¼ 1; 2 refer to the lower and upper walls, respectively.
(15)
C2 ¼ C5 þ C6 C1 þ
3
2ð3F1 1Þ2 i þ ð4 þ F3 Þð1 2F2 Þ ;
h
6 F1 F3 F22
146
G. Ibáñez et al. / Energy 50 (2013) 143e149
3F4 C C3 ¼ þ 1; 2g1 ð3F1 1Þ2 g1
4.1455
a2 0 a 2 0.01
4.145
a 2 0.015
C C4 ¼ 3 ½1 þ Bi1 ðd1 þ 1Þ þ Qa ; Bi1 3F5
4.1445
S 4.144
C1
þ ; C5 ¼ 2g2 ð3F1 1Þ2 g2
4.1435 4.143
C C6 ¼ 5 ½1 þ Bi2 ðd2 þ 1Þ þ Qa ; Bi2
0
0.02
0.04
0.06
0.08
0.1
0.12
a1
with
F3 ¼ APe
Fig. 1. Global entropy generation rate as a function of the upper wall slip length, a1 , for different values of a2 . Pe ¼ 0:1, A ¼ 7:8, g1 ¼ g2 ¼ 12, Bi1 ¼ Bi2 ¼ 2 and d1 ¼ d2 ¼ 0:05.
3F1 1 ; 3
F4 ¼ 6 F1 F3 F22 ð4 þ F3 Þð2 þ 3F2 Þ; F5 ¼ 6 F1 F3 F22 þ ð4 þ F3 Þð2 3F2 Þ: Although not shown here, the temperature of the system decreases with the slip length since the fluid friction decreases. Moreover, the temperature of the system also decreases with both the thermal conductivity of the walls and the Biot numbers since the heat transfer to the surroundings is higher; however, when the thickness of the walls is high the temperature of the system increases since the thermal resistance of the walls is higher. Consequently, the heat transfer to the ambient is lower. 3. Results 3.1. Entropy generation The velocity and temperature fields already obtained will be used for the determination of the entropy generation rate within the microchannel. In the flow of a viscous, incompressible and monocomponent fluid, the local entropy generation rate, can be written explicitly in dimensionless terms as [49,51]
2 2 vQ vQ þ vx vy Q 2 g vQw1 g vQw2 2 1 du 2 þ 22 þ ; þ 21 Q dy vy vy Q Q
1 S_ ¼ 2
w1
(18)
w2
where S_ is normalized by k=a2. In Eq. (18), the first three terms account for irreversibilities caused by heat flow in the fluid and walls while the fourth terms express the irreversibilities due to viscous dissipation in the fluid, respectively. In the first term, the entropy generation by heat transfer in the fluid due to both axial and transverse heat conduction is considered. The global entropy _ is once more obtained by integrating S_ in the generation rate, hSi, whole volume occupied by the microdevice. Once integrated, this quantity only depends on the dimensionless parameters A, Pe, gi , Bii , di , ai and Qa that govern the performance of the system. In all cases, the dimensionless ambient temperature has been fixed to Qa ¼ 5 using the physical properties of engine oil [52] at an ambient temperature Ta ¼ 20 C. Since the global entropy generation rate considers the whole dissipation produced by irreversibilities in the system, we can look for values of the parameters that minimize this function. In Fig. 1 the global entropy generation rate is reported as a function of the upper wall slip length, a1 , for three
different values of lower wall slip length, a2 . Pe ¼ 0:1, A ¼ 7:8, g1 ¼ g2 ¼ 12, Bi1 ¼ Bi2 ¼ 2 and d1 ¼ d2 ¼ 0:05. We observe, that the global entropy generation decreases with a1 , reaching a minimum for all values of a2 and then exhibits an increase. In order to understand the nature of this behavior, the independent contributions to the global entropy generation in the system associated to both, heat conduction in the fluid and solid walls and viscous dissipation in the fluid are analyzed. The entropy generation produced by heat transfer in the fluid always increases with a1 due to the decrease in the temperature, which produce an increase in the entropy generation due to its inverse dependence; however, the terms associated to heat transfer in the lower wall and viscous dissipation in the fluid always decrease with a1 due to the reduction in the lower wall temperature gradient and velocity gradient, respectively. On the other hand, the entropy generation associated to heat transfer in the upper wall decreases for small values of a1 and reaches a minimum value. Once the minimum value is reached, this term exhibits an increase as a1 grows. For small values of a1 , the reduction in the irreversibilities associated to viscous dissipation and heat transfer in the lower and upper walls dominates over the increment in the term associated to heat transfer in the fluid in _ shows a minimum value. Once the minimum such a way that hSi value is reached, the increment in term associated to heat transfer _ exhibits an increase as a grows. In in the fluid dominates and hSi 1 addition, it is found that the major contribution corresponds to irreversibilities due to heat transfer in the fluid. This term increases with a2 at constant a1 , therefore in Fig. 1 the higher the lower wall slip length, a2 , the higher the value of entropy where the minimum is reached. On the other hand, when a2 increases, the increment in the term associated to heat transfer in the fluid dominates over the other terms at lower values of a1 , therefore the higher the value of a2 the lower the optimum value of a1 where the minimum is reached. In Fig. 2 the global entropy generation rate is reported as a function of the single slip length (symmetric slip conditions), a, for three different values of d1 ¼ d2 , and Pe ¼ 0:1, A ¼ 7:8, g1 ¼ g2 ¼ 12 and Bi1 ¼ Bi2 ¼ 2. Once more, the entropy generation reaches minimum values. These minima move to higher values of a when d1 ¼ d2 decreases. Fig. 3 shows the normalized global entropy generation rate as a function of the Biot number of the lower wall, Bi1 , for different values of a1 ¼ a2 . The other parameters are Pe ¼ 0:1, A ¼ 1, g1 ¼ g2 ¼ 2, Bi2 ¼ 10 and d1 ¼ d2 ¼ 0:05. As can be observed from Fig. 3, for this case the global entropy generation rate presents a minimum value for the three different values of a1 ¼ a2 . It is possible to find an optimum Biot number for the lower surface, Bi1opt , which leads to _ Similarly to what a minimum global entropy generation rate hSi.
G. Ibáñez et al. / Energy 50 (2013) 143e149
4.18
4.175
d1 d2 0.02
1
Bi 2,
0
d1 d2 0.05
0.99
Bi 2,
0.05
0.98
Bi 3,
0.05
d1 d2 0.08
S
S
147
S
4.17
g1
0.97
0
0.96 0.95
4.165
0.94 0
4.16 0
0.02
0.04
0.06
a Fig. 2. Global entropy generation rate as a function of the single slip length, a, for different values of d1 ¼ d2 . Pe ¼ 0:1, A ¼ 7:8, g1 ¼ g2 ¼ 12, Bi1 ¼ Bi2 ¼ 2 (symmetric slip condition).
occurred in the non conjugate heat transfer problem [53,54], we found that in the present example the global entropy generation rate is always a monotonous increasing function of Bi and reaches a limiting value as Bi/N, when the Biot numbers of each wall are the same. Fig. 4 shows the global entropy generation rate as a function of the lower wall thermal conductivity, g1 , normalized by its value when g1 ¼ 0 for different values of a1 ¼ a2 and Bi1 ¼ Bi2 , i.e. symmetric conditions of slip and convective cooling, respectively. The other parameters are Pe ¼ 0:1, A ¼ 1, g2 ¼ 10 and d1 ¼ d2 ¼ 0:1. Notice that eventhough the cooling of the walls _ presents minimum values for the three values of is symmetric, hSi Bi1 ¼ Bi2 and a1 ¼ a2 explored. This means that for a given fluid, irreversibilities can be minimized by an adequate selection of the thermal conductivities of the walls. The optimum thermal conductivity, g1opt , has higher values with both decreasing values of a and Bi, respectively. Figs. 5 and 6, show the effects of both the axial temperature gradient, A, and dimensionless Péclet number, Pe, on the global entropy generation rate for three different values of a1 ¼ a2 . Fig. 5 refers to Pe ¼ 0:1, g1 ¼ g2 ¼ 2, Bi1 ¼ Bi2 ¼ 4 and d1 ¼ d2 ¼ 0:1, while Fig. 6 refers to A ¼ 1, g1 ¼ g2 ¼ 10, _ Bi1 ¼ Bi2 ¼ 3 and d1 ¼ d2 ¼ 0:1. In both, Figs. 5 and 6, hSi decreases for small values of A and Pe, respectively, and reaches _ exhibits a minimum value. Once the minimum value is reached, hSi an increase as A and Pe grows. This mean that there is an optimum value of A and Pe, that minimizes the irreversibilities associated to this system, provided the other parameters remain fixed. These optimum values of A and Pe, where the entropy generation is minimum, decrease with the slip length. In Fig. 5, the smallest
0.8
entropy generation is obtained at the highest value of a1 ¼ a2 . Although not shown here, the minima of the global entropy generation in Fig. 6 move to lower values of Péclet number when d1 ¼ d2 increases. Moreover, when we consider the case where the Biot numbers for each wall are equal, i.e. conditions of symmetric convective cooling, it is possible to find again an optimum Péclet number which leads to a minimum global entropy generation rate. 3.2. Nusselt number The local Nusselt number at the upper wall is given by [49]
dQ dy y¼1 ; ¼ 2 Qy¼1 Qb
Nu ¼
k vT Twðy0 ¼ aÞ Tb vy0
y0
(19)
where Tb and Twðy0 ¼ aÞ are the dimensional expressions of the bulk temperature (i.e. the cross-section averaged temperature of the stream) and the fluidewall interface temperature at y0 ¼ a, respectively. The dimensionless bulk temperature is defined as
Z1 uQdy
Qb ¼
1
:
Z1
(20)
udy 1
1.8
a1
2
0
1.75
a1
2
0.01
a1
2
0.02
1.7
0.98
Bi1
0.6
Fig. 4. Normalized global entropy generation rate as a function of the lower wall thermal conductivity, g1 , for different values of Bi1 ¼ Bi2 ¼ and a1 ¼ a2 . Pe ¼ 0:1, A ¼ 1, g2 ¼ 10 and d1 ¼ d2 ¼ 0:1.
1
S
0.4
g1
1.02
S
0.2
0.08
S 1.65
0.96 0
1.6
0.94
a1 a1 a1
0.92 0
0.2
0.4
0.6
0.8
1
1.2
2 2 2
0 0.05 0.1 1.4
Bi1 Fig. 3. Normalized global entropy generation rate as a function of the Biot number of the lower wall, Bi1 , for different values of a1 ¼ a2 . Pe ¼ 0:1, A ¼ 1, g1 ¼ g2 ¼ 2, Bi2 ¼ 10 and d1 ¼ d2 ¼ 0:05.
1.55 0
0.5
1
1.5
2
2.5
A Fig. 5. Global entropy generation rate as a function of the axial temperature gradient, A, for different values of a1 ¼ a2 . Pe ¼ 0:1, g1 ¼ g2 ¼ 2, Bi1 ¼ Bi2 ¼ 4 and d1 ¼ d2 ¼ 0:1.
148
G. Ibáñez et al. / Energy 50 (2013) 143e149
1.002
1.04
a1 a1 a1
1.001
2 2 2
0 0.01 0.02
1.03
1 1.02
0.999
d1 d1
1.01
d 2 0.05 d 1 d 2 0.075
0.998 0
0.05
0.1
0.15
0.2
0.25
0.3
Pe
Nusselt number as a function of the upper wall slip length, a1 , for different values of the lower wall slip length, a2 , and Pe ¼ 0:1, A ¼ 10, g1 ¼ 1, g2 ¼ 0:2, Bi1 ¼ Bi2 ¼ 2 and d1 ¼ d2 ¼ 0:05 is presented in Fig. 7. Nusselt number have been normalized by its value when a1 ¼ 0. This local Nusselt number displays a maximum value, namely, there is an optimum value of the dimensionless upper wall slip length, a1 , where the heat transfer is maximum. Also, the higher the value of a2 , the lower the optimum value of a1 where the heat transfer is maximum. Fig. 8 shows the normalized Nusselt number as a function of the upper wall slip length, a1 , for different values of the single wall thickness d1 ¼ d2 , and Pe ¼ 0:1, A ¼ 10, g1 ¼ 1, g2 ¼ 0:2, Bi1 ¼ Bi2 ¼ 2 and a2 ¼ 0:05. The normalized Nusselt number reaches maximum values that become higher as d1 ¼ d2 increases. The optimum value of a1 , where the heat transfer is maximum, increases with the wall thickness. Two different optimization objectives have been considered in the analysis, namely, minimization of entropy generation and maximization of heat transfer coefficient (Nu). These two functions are not optimized simultaneously because in the present work, our fundamental purpose is to show that there are optimum values of the slip length that minimize the entropy production and there are optimum values of the slip length that maximize the heat transfer, separately. In addition, minimum entropy generation and maximum heat transfer can be two different optimization criteria which can be selected according to the application for a particular case. For example, generally the well known entropy generation minimization method is valid for optimizing thermal processes
0.79 a 2 0.065
0.78
a 2 0.07
0.77
a 2 0.075
Nu
0.76 0.75 0.74 0.73 0
0.05
0.1
0.15
1 0
Fig. 6. Normalized global entropy generation rate as a function of the Péclet number, Pe, for different values of a1 ¼ a2 . A ¼ 1, g1 ¼ g2 ¼ 10, Bi1 ¼ Bi2 ¼ 3 and d1 ¼ d2 ¼ 0:1.
0.2
0.25
a1 Fig. 7. Nusselt number as a function of the upper wall slip length, a1 , for different values of the lower wall slip length a2 . Pe ¼ 0:1, A ¼ 10, g1 ¼ 1, g2 ¼ 0:2, Bi1 ¼ Bi2 ¼ 2 and d1 ¼ d2 ¼ 0:05.
d 2 0.025
0.05
0.1
0.15
0.2
a1 Fig. 8. Normalized Nusselt number as a function of the upper wall slip length, a1 , for different values of the wall thickness d1 ¼ d2 . Pe ¼ 0:1, A ¼ 10, g1 ¼ 1, g2 ¼ 0:2, Bi1 ¼ Bi2 ¼ 2 and a2 ¼ 0:05.
where heat-work conversion is presented, while the heat transfer maximization is more suitable in heat exchanger optimizations involving only heat transfer processes. In particular, Chen et al. [24] showed that the minimum entropy generation principle is valid for optimizing heat exchangers in a thermodynamic cycle while the entransy dissipation maximization is more suitable in heat exchanger optimizations involving only heat transfer processes. However, the problem of the present paper will be investigated using a multi-objective optimization technique to maximize the heat transfer and to minimize the entropy production, simultaneously. On the other hand, although the best combination of all parameters simultaneously is possible. For example, using Lagrange multipliers technique for finding the extrema of a multivariable function. In this work, we optimize the system using individual parameter with some other parameters fixed because in real design situations of thermal systems is necessary to fix some parameters according to the real conditions. Moreover, although the results in Figs. 1e6 are for a particular choice for some parameters, we have verified that a minimum of the entropy production exists in a large range of values of these parameters. 4. Summary and conclusion In this work we applied the entropy generation minimization method to the optimization of a viscous flow that takes place in a parallel plate microchannel with walls of finite thickness. The velocity field was determined using slip boundary conditions. The temperature field was determined from the conjugate heat transfer problem considering convective heat transfer and assuming a thermally fully developed flow. All relevant design parameters for the system, including geometric parameters, material properties and flow conditions were optimized by minimizing the entropy generation rate. We have shown that the system can operate under minimum global entropy generation conditions that correspond to minimize irreversibilities due to viscous friction and heat flow in the fluid and walls. In particular, an optimum slip velocity which leads to a minimum global entropy generation rate was obtained and the effects of slip velocity on the optimum values of some other parameters were analyzed. The Nusselt number, that is, the dimensionless heat transfer coefficient at the walls of the microchannel, was also calculated and analyzed in some specific cases. An optimum value of the slip length that maximizes the heat transfer was derived. The results of the present study provide valuable fundamental information for the design and optimization of heat transfer equipment.
G. Ibáñez et al. / Energy 50 (2013) 143e149
Acknowledgments G. Ibáñez thankfully acknowledges financial support from PROMEP.
References [1] Bejan A. Second law analysis in heat transfer. Energy 1980;5(8e9):720e32. [2] Bejan A. Minimization of entropy generation. Boca Raton: CRC Press; 1996. [3] San JY. Second-law performance of heat exchangers for waste heat recovery. Energy 2010;35(5):1936e45. [4] Revellin R, Lips S, Khandekar S, Jocelyn Bonjour J. Local entropy generation for saturated two-phase flow. Energy 2009;34(9):1113e21. [5] Sciacovelli A, Verda V. Entropy generation analysis in a monolithic-type solid oxide fuel cell (SOFC). Energy 2009;34(7):850e65. [6] Li X, Faghri A. Local entropy generation analysis on passive high concentration DMFCs (direct methanol fuel cell) with different cell structures. Energy 2011; 36:403e14. [7] Rangel VH, Damian C, Juarez D, Gallegos A, Zaleta A, Plascencia H. Entropy generation analysis of a proton exchange membrane fuel cell (PEMFC) with a fermat spiral as a flow distributor. Energy 2011;36:4864e70. [8] Ibáñez G, López de Haro M, Cuevas S. Thermodynamic optimization of radial MHD flow between parallel disks. J Non-Equilib Thermodyn 2004;29:107e22. [9] Ibáñez G, Cuevas S, López de Haro M. Optimization analysis of an alternate MHD electric generator. Energy Convers Manag 2002;43(14):1757e71. [10] Ibáñez G, Cuevas S, López de Haro M. Optimization of a magnetohydrodynamic flow based on the entropy generation minimization method. Int Comm Heat Mass Transfer 2006;33:295e301. [11] Ibáñez G, Cuevas S. Optimum wall conductance ratio in magnetoconvective flow in a long vertical rectangular duct. Int J Thermal Sci 2008;47:1012e9. [12] Saidi MH, Montazeri A. Second law analysis of a magnetohydrodynamic plasma generator. Energy 2007;32(9):1603e16. [13] Makinde OD, Aziz A. Analysis of entropy generation and thermal stability in a long hollow cylinder with asymmetry convective cooling. Heat Mass Transfer 2011;47(11):1407e15. [14] Aziz A, Khan WA. Classical and minimum entropy generation analyses for stead-381 y state conduction with temperature dependent thermal conductivity and asymmetric thermal boundary conditions: regular and functionally graded materials. Energy 2011;36:6195e207. [15] Makinde OD. Irreversibility analysis of variable viscosity channel flow with convective cooling at the walls. Can J Phys 2008;86(2):383e9. [16] Tshehla MS, Makinde OD, Okecha GE. Heat transfer and entropy generation in a pipe flow with temperature dependent viscosity and convective cooling. Scientific Res Essays 2010;5(23):3730e41. [17] Basak T, Anandalakshmi R, Kumar P, Roy S. Entropy generation vs energy flow due to natural convection in a trapezoidal cavity with isothermal and nonisothermal hot bottom wall. Energy 2012;37:514e32. [18] Kaluri RS, Basak T. Entropy generation due to natural convection in discretely heated porous square cavities. Energy 2011;36:5065e80. [19] Chen S, Du R. Entropy generation of turbulent double-diffusive natural convection in a rectangle cavity. Energy 2011;36:1721e34. [20] Mahian O, Mahmud S, Zeinali Heris S. Analysis of entropy generation between co-rotating cylinders using nanofluids. Energy 2012;44:438e46. [21] Ahmadi P, Hajabdollahi H, Dincer I. Cost and entropy generation minimization of a cross-flow plate fin heat exchanger using multi-objective genetic algorithm. Trans ASME, J Heat Transfer 2011;133:021801 (1e10). [22] Guo ZY, Zhu H, Liang XG. Entransy-A physical quantity describing heat transfer ability. Int J Heat Mass Transfer 2007;50:2545e56. [23] Chen Q, Zhu H, Pan N, Guo ZY. An alternative criterion in heat transfer optimization. Proc R Soc A 2011;467:1012e28. [24] Chen Q, Wu J, Wang M, Pan N, Guo ZY. A comparison of optimization theories for energy conservation in heat exchanger groups. Chin Sci Bull 2011;56(4e 5):449e54. [25] Chen Q, Wang M, Pan N, Guo ZY. Optimization principles for convective heat transfer. Energy 2009;34:1199e206. [26] Chen Q, Xu YC. An entransy dissipation-based optimization principle for building central chilled water systems. Energy 2012;37:571e9.
149
[27] Haddad O, Abuzaid M, Al-Nimr M. Entropy generation due to laminar incompressible forced convection flow through parallel-plates microchannel. Entropy 2004;6(5):413e26. [28] Chen K. Second-law analysis and optimization of microchannel flows subjected to different thermal boundary conditions. Int J Energy Res 2005;29(3): 249e63. [29] Abbassi H. Entropy generation analysis in a uniformly heated microchannel heat sink. Energy 2007;32(10):1932e47. [30] Erbay LB, Yalcln MM, Ercan MS. Entropy generation in parallel plate microchannels. Heat Mass Transfer 2007;43:729e39. [31] Guo J, Xu M, Cai J, Huai X. Viscous dissipation effects on entropy generation in curved square microchannels. Energy 2011;36:5416e23. [32] Singh PK, Anoop KB, Sundararajan T, Das SK. Entropy generation due to flow and heat transfer in nanofluids. Int J Heat Mass Transfer 2010;53: 4757e67. [33] Li J, Kleinstreuer C. Entropy generation analysis for nanofluid flow in microchannels. J Heat Transfer 2010;132:122401e11. [34] Naterer GF, Adeyinka OB. Microfluidic exergy loss in a non-polarized thermomagnetic field. Int J Heat Mass Transfer 2005;48:3945e56. [35] Glockner PS, Naterer GF. Numerical simulation of electrokinetic flow and heat transfer in microchannels with a finite-volume method. Num Heat Transfer Part A 2006;49:451e70. [36] Naterer GF. Microfluidic friction and thermal energy exchange in a nonpolarized electromagnetic field. Int J Energy Res 2007;31:728e41. [37] Naterer GF. Effects of magnetic field strength and Hartmann number on electro-dynamic energy exchange in microchannels. In: Proc. 38th AIAA plasmadynamics and lasers conference, vol. 2; 2007. p. 1132e9. [38] Ibáñez G, Cuevas S. Entropy generation minimization of a MHD (magnetohydrodynamic) flow in a microchannel. Energy 2010;35:4149e55. [39] Shamshiri M, Khazaeli R, Ashrafizaadeh M, Mortazavi S. Heat transfer and entropy generation analyses associated with mixed electrokinetically induced and pressure-driven power-law microflows. Energy 2012;42:157e69. [40] Linga Raju T. Magnetohydrodynamic slip-flow regime in a rotating channel. J Ind Acad Math 2007;29:27. [41] Makinde OD, Osalusi E. MHD steady flow in a channel with slip at the permeable boundaries. Rom J Phys 2006;51:319. [42] Smolentsev S. MHD duct flows under hydrodynamic slip condition. Theo Comput Fluid Dyn 2009;23:557. [43] Yu Shiping, Ameel Timothy A. Slip-flow heat transfer in rectangular microchannels. Int J Heat Mass Transfer 2001;44:4225e34. [44] Hooman K. A superposition approach to study convection in straight microchannels of uniform but arbitrary cross-section. Int J Heat Mass Transfer 2008; 51:3753e62. [45] Ogedengbe EOB, Naterer GF, Rosen MA. Slip flow irreversibility of dissipative kinetic and internal energy exchange in microchannels. J Micromech Microeng 2006;16:2167e76. [46] Ogedengbe EOB, Naterer GF, Rosen MA. Slip flow irreversibility effects and conjugate heat transfer in a counterflow heated microchannel. In: 3rd IASME/ WSEAS Int. Conf. on energy and environment. UK: University of Cambridge; February 2008. p. 23e5. [47] Hooman K. Heat transfer and entropy generation for forced convection through a microduct of rectangular cross-section: effects of velocity slip, temperature jump, and duct geometry. Int Comm Heat Mass Transfer 2008; 35(9):1065e8. [48] Yazdi MH, Abdullah S, Hashim I, Sopian K, Zaharim A. Entropy generation analysis of liquid fluid past embedded open parallel microchannels within the surface. Eur J Sc Res 2009;28(3):462e70. [49] Ibáñez G, López A, Cuevas S. Optimum wall thickness ratio based on the minimization of entropy generation in a viscous flow between parallel plates. Int Comm Heat Mass Transfer 2012;39:587e92. [50] Snyder WT. The influence of wall conductance on magnetohydrodynamic channel-flow heat transfer. J Heat Transfer Trans ASME 1964:552e8. [51] de Groot SR, Mazur P. Non-equilibrium thermodynamics; 1984. Dover, New York. [52] Holman JP. Heat transfer. 7th ed. New York: Mc Graw-Hill; 1990. [53] Ibáñez G, Cuevas S, López de Haro M. Heat transfer in asymmetric convective cooling and optimized entropy generation rate. Rev Mex Física 2003;49: 338e43. [54] Ibáñez G, Cuevas S, López de Haro M. Minimization of entropy generation by asymmetric convective cooling. Int J Heat Mass Transfer 2003;46:1321e8.
