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Entropy generation and natural convection in rectangular cavities
Applied Thermal Engineering 29 (2009) 1417–1425
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Entropy generation and natural convection in rectangular cavities Rejane De C. Oliveski *, Mario H. Macagnan, Jacqueline B. Copetti Department of Mechanical Engineering, School of Physical and Technological Sciences, Universidade do Vale do Rio dos Sinos, UNISINOS, Av. Unisinos 950 São Leopoldo, CEP 93020-000, RS, Brazil
a r t i c l e
i n f o
Article history: Received 14 September 2006 Accepted 15 July 2008 Available online 23 July 2008 Keywords: Entropy Natural convection Bejan number Rayleigh number
a b s t r a c t This work presents a numerical analysis of entropy generation in rectangular cavities that were submitted to the natural convection process. This natural convection process was caused by temperature differences between the vertical walls of the cavities. Momentum and energy equations were used to solve this problem. These equations were coupled by the Boussinesq approximation. Initially the cavities were submitted to uniform temperature and velocity fields. The hypothesis of perfect insulation was considered for the top and bottom walls of the cavity. Impermeability and non-slip condition in the boundary were assumed for every wall of the cavity. The numerical analysis is performed through a two-dimensional model with the Finite Volume method. The results of the entropy generation obtained to a square cavity were used to validate the numerical model and it presented good concordance with results from other authors. Additionally, an analysis of the entropy generation in rectangular cavities was performed with five aspect ratios, five Rayleigh numbers and four irreversibility coefficients. The results of this work indicate that: (a) the total entropy generation in steady state increases linearly in both cases, the aspect ratio and the irreversibility coefficient, and exponentially with the Rayleigh number; (b) the influence of the aspect ratio on Bejan number is proportional to Rayleigh number and inversely proportional to the irreversibility coefficient; (c) for the same aspect ratio, the entropy generation due to the viscous effects increases with the Rayleigh number and, for a certain Rayleigh number, the entropy generation due to the viscous effects also increases with the aspect ratio. Ó 2008 Elsevier Ltd. All rights reserved.
Contents 1. 2.
3.
4.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mathematical model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. Initial and boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. Entropy generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3. Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Numerical validation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1. Entropy generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.2. Nusselt number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Case study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1. Introduction Natural convection heat transfer is present in some important processes in engineering such as thermal storage, environmental * Corresponding author. Tel.: +55 51 3591 1100x1719. E-mail address: [email protected] (R. De C. Oliveski). 1359-4311/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.applthermaleng.2008.07.012
1417 1418 1419 1419 1420 1420 1420 1420 1421 1421 1423 1424 1424
comfort, grain drying, electronic cooling and others. Whatever the shape of the wall, flow and heat transfer problems inside enclosures have numerous engineering applications like solar-collectors, double-wall insulation, electric machinery, cooling system of electronic devices, natural circulation in the atmosphere, etc. [1]. In natural convection processes, the thermal and the hydrodynamic are coupled and both are, according to Bejan [2], strongly
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Nomenclature A Be Cp g h H k L Nu p P Pr Ra S_ t T u, v U, V x, y X, Y
aspect ratio, H/L Bejan number, S_ l;a;h =S_ l;a specific heat at constant pressure (J/kg K) gravitational acceleration (m/s2) heat transfer coefficient (W/m2 K) cavity height (m) thermal conductivity (W/m K) cavity length (m) Nusselt number, hL/k pressure (Pa) dimensionless pressure, pL2/qa2 Prandtl number, t/a Rayleigh number, (gbDTL3)/(at) entropy generation (W/m3 K) time (s) temperature (°C) velocity components in x, y direction (m/s) dimensionless velocity, components in x, y direction, uL/a, mL/a cartesian coordinates (m) dimensionless cartesian coordinates, x/L, y/L
influenced by the fluid thermophysical characteristics, the temperature differences and the system geometry. The existence of a large number of systems submitted to the natural convection process, involving different physical and constructive characteristics, makes the study of this phenomenon an always present theme, despite the several decades of research on the subject. Several works about natural convection in many different areas are available, with experimental [3–5] and numerical approaches [6–11]. All real processes present irreversibilities and can be associated to friction, mass transference, thermal gradients, chemical reactions, etc., resulting in process efficiency loss. This efficiency loss is related to the entropy generation. As the generation of entropy destroys the system energy, its minimization has been used as the optimal design criteria for thermal systems. The mechanisms of entropy generation shall be understood [12,13] and such understanding can be achieved with a detailed study of the process, with the analysis of this dependence with the different thermal, hydrodynamic and geometric characteristics of these processes. In this way, an optimal configuration with minimum loss of available energy may be obtained. Many entropy generation works about flows with heat transfer, in both case, natural and forced convection regime, can be found in the literature. In the case of forced convection regime it can be mentioned the works of Haddad et al. [11], Khalkhali et al. [14], Sahin [15,16] and Abbassi et al. [17], among others. Concerning entropy generation in natural convection regime it can be mentioned the works of Zahmatkesh [8], Abu-Hijleh et al. [18], Abu-Hijleh and Heilen [19], Mahmud and Island [1], Magherbi et al. [20] and others. Sahin [15] analyzed the variation of entropy generation in function of viscosity in forced flow. For this problem, the authors prescribed a constant temperature to the duct walls. In another work, the same author [16] made a similar analysis with constant heat flow imposed to the duct wall. Mahmud and Fraser [21] developed analytical solutions for entropy generation, irreversibility distribution and Bejan number for a series of classical forced convection problems, from their analytic solution of velocity and temperature profile. For a range of Rayleigh numbers and irrever-
Greeks
a b
l m q f h u
thermal diffusivity (m2/s) thermal expansion coefficient (K1) dynamic viscosity (kg/ms) kinematic viscosity (m2/s) density (kg/m3) dimensionless time, ta/L2 dimensionless temperature, (T Tc)/(Th Tc) irreversibility ratio, (lT0/k)(a/L(Th Tc))2
Subscripts a dimensionless c could f fluid viscous effect h hot, heat transfer l local max maximum p permanent T total
sibilities, Abbassi et al. [17] investigated numerically the entropy generation in Poiseuille–Bernard flows, where Reynolds and Peclet numbers were kept constant. The entropy generation due to natural convection in a horizontal cylinder was investigated numerically by Abu-Hijleh et al. [18]. In this work, the authors showed that, for a range of Rayleigh number, the entropy generation decreases according to the increase of the cylinder diameter. They also showed that the inverse happens when the Rayleigh number is increased in relation to the same diameter. Recently, Zahmatkesh [8] analyzed the importance of thermal boundary conditions in heat transfer process and entropy generation characteristics inside of a porous enclosure. To do this, the author simulated the natural convection processes in a porous enclosure over a wide range of Darcy-modified Rayleigh number. Zahmatkesh [8] concluded that the generation rate is likely to be the highest for uniform heating/cooling and the lowest for non-uniform heating and that the optimum case with respect to heat transfer and entropy generation could be achieved by non-uniform heating. The aim of the present work is the study of entropy generation in natural convection processes in rectangular cavity through numerical simulation. The cavity was submitted to a horizontal temperature gradient, inducing the natural convection process inside the cavity. These conditions could be applied for solar thermal collectors and reservoirs, cooling in electronic enclosures and others. Initially, a numerical validation is presented, comparing some results of this work with that of entropy generation found in the literature for a square cavity. The contribution of this work is the analyses of the variation of entropy generation in relation to Rayleigh number, aspect ratio and irreversibility coefficient in rectangular cavities submitted to natural convection process, allowing identifying the source and location of entropy generation. 2. Mathematical model To solve the entropy generation problem due to natural convection in rectangular cavities, it is assumed that: the fluid is incompressible, the Boussinesq approximation is valid and that the flow has two-dimensional characteristics, as indicated in Fig. 1.
R. De C. Oliveski et al. / Applied Thermal Engineering 29 (2009) 1417–1425
y
oV o o oP o oV þ ðUVÞ þ ðVVÞ ¼ þ Pr of oX oY oY oX oX o oV þ Pr þ Pr Rah; oY oY oh o o o oh o oh þ ; þ ðUhÞ þ ðVhÞ ¼ of oX oY oX oX oY oY
∂T ∂y = 0
Th
Tc
g
u x
L
2.2. Entropy generation
Fig. 1. Schematic view of cavity.
Furthermore, it is assumed that the fluid in the cavities is Newtonian and has constant properties. In this case, the governing equations are the ones that follow:
o o ð1Þ ðquÞ þ ðqvÞ ¼ 0; ox oy o o o op o ou o ou l l ðquÞ þ ðquuÞ þ ðqvuÞ ¼ þ þ ; ot ox oy ox ox ox oy oy
ð3Þ ð4Þ
where u and v are, respectively, the velocities in ‘‘x” and ‘‘y” directions, q is the density, p is the pressure, l is the absolute viscosity, and b is the fluid thermal expansion coefficient. The variable a represents the thermal diffusivity, T, the temperature, t, time, and g, gravity acceleration. The dimensionless form of the Eqs. (1)–(4) can be obtained by applying the following parameters:
h¼
at
T Tc ; Th Tc
;
Pr ¼
m ta H ; f¼ 2; A¼ ; L a L
ðX; YÞ ¼
x y ; ; L L
ðU; VÞ ¼
uL vL ; ;
a a
In the natural convection process the entropy generation is associated to the heat transfer and to the fluid flow friction. According to Bejan [22], the local entropy generation (S_ l ) can be determined by the following expression:
2 2 # oT oT þ ox oy " 2 2 # 2 l ou ov ou ov þ ; 2 þ2 þ þ ox oy oy ox T0
k S_ l ¼ 2 T0
ð2Þ
o o o op o ov l ðqvÞ þ ðquvÞ þ ðqvvÞ ¼ þ ot ox oy oy ox ox o ov þ l þ qbgðT T C Þ; oy oy oT o o o oT o oT a a þ ðuTÞ þ ðvTÞ ¼ þ ; ot ox oy ox ox oy oy
gbDTL3
ð9Þ
The boundary conditions for the hydrodynamic problem are impermeability and non-slipping in all cavity walls. For the thermal problem, the horizontal walls were kept adiabatic while the vertical walls were kept at a constant temperature. The hot wall was kept with h = 0.5 and cold wall with h = 0.5. In all domain the initial conditions were: zero to the velocity field and h = 0.5 X to the temperature field.
v
Ra ¼
ð8Þ
2.1. Initial and boundary conditions
H
∂T ∂y = 0
1419
P¼
pL2
qa
2
;
where Ra and Pr are Rayleigh and Prandtl numbers, respectively, v is the kinematic viscosity, Tc is the cold wall temperature, Th is the hot wall temperature and h is the dimensionless temperature. The time is represented by t and the dimensionless time by f, the pressure by p and the dimensionless pressure by P. Besides these variables, A is the aspect cavity ratio, U and V are the dimensionless velocities in X and Y directions, respectively, also dimensionless. After the replacement of this parameters in Eqs. (1)–(4) it can be obtained the dimensionless equations of the problem (Eqs. (6)– (9)), as presented in [20].
ð6Þ ð7Þ
ð10Þ
where k is the thermal conductivity of the fluid. The first term of Eq. (10) represents the dimensional entropy generation due to heat transfer (S_ l;h ) while the second term represents the entropy generation due to the viscous effects of the fluid (S_ l;f ), so that
S_ ¼ S_ l;h þ S_ l;f :
ð11Þ
This equation can be rewritten by using the dimensionless parameters showed in Eq. (5), that results in the following expressions:
2 2 # oh oh ; þ oX oY " 2 2 # 2 oU oV oU oV ; ¼u 2 þ2 þ þ oX oY oY oX
S_ l;a;h ¼ S_ l;a;f
"
S_ l;a ¼ S_ l;a;h þ S_ l;a;f ; 2 lT a u¼ 0 ; k LðT h T c Þ ð5Þ
oU oV þ ¼ 0; oX oY oU oðUU Þ oðVV Þ oP o oU o oU þ þ ¼ þ Pr þ Pr ; of oX oY oX oX oX oY oY
"
ð12Þ ð13Þ ð14Þ
Th þ Tc T0 ¼ ; 2
ð15Þ
where T0 is the bulk temperature and u is the ratio between the viscous and thermal irreversibilities. The dimensionless total entropy generation (S_ t;a ) is obtained through the integration of the local entropy generation (S_ l;a ) in all computational domain, as indicated in Eq. (16).
S_ T; a ¼
Z
S_ l;a dv:
ð16Þ
v
An alternative parameter for irreversibilities distribution is the Bejan number (Be) defined as [20]
Be ¼
S_ l;a;h : S_ l;a
ð17Þ
The heat transfer irreversibilities are dominant when Be 1/2. When Be 1/2 the irreversibilities due to the viscous effects dominate the processes and if Be = 1/2 the entropy generation due to the viscous effects and the heat transfer are equal [20].
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2.3. Methodology The numerical solution is obtained by the discretization of Eqs. (6)–(9), in Finite Volume method, as described by Pantakar [23]. The pressure–velocity coupling is obtained by the SIMPLEC method from Van Doormaal and Raithby [24]. The Power Law scheme is used for interpolation in the faces of the control volumes. The resulting equation systems are solved by the TDMA algorithm. The convergence criterion used was the mass conservation residue, established in 108. It was used a spatial grid non-uniformly spaced, with 70 70 volume elements, with volume concentration in the cavities walls. For that, an equation for volume distribution suggested by Davidson [25] it was used, with a grid refinement coefficient equal to 2.0. 3. Results Two types of analysis are presented in this work. The first on represents the validation of the mathematical model and its implementation (Section 3.1). The second shows the contribution of this work in relation of the entropy generation in natural convection process (Section 3.2). 3.1. Numerical validation
tion and the curve behavior are different. In this figure, it can be also observed that, after the transient period, the entropy generation remains practically constant. On the other hand, in Fig. 2b, it can be observed that after a high increase of entropy generation, it decreases significantly, with further oscillations before reaching the steady state regime. There are some explanations in the literature [19,20] for this fact. Magherbi et al. [20] says that for small Rayleigh numbers the steady state is very close to the equilibrium state and therefore the system reaches the steady state regime rapidly, without passing through the oscillation period. On the other hand, for Ra > Rac = 5200 the steady state regime draws away from the equilibrium state. In this case, oscillations are observed around the stationary state, which correspond to the total entropy generation oscillations. By definition, if Bejan number is equal to 1/2 it means that the thermal and viscous irreversibilities are equivalent. Above this value the thermal irreversibilities predominate. In Fig. 3, Bejan number is presented in function of the dimensionless time, which was obtained with Eq. (17) for Ra = 104 and four values of irreversibility distribution. These results are compared to that one that was obtained by Magherbi et al. [20]. In Fig. 3, it can be observed that for u = 104 (Fig. 3c) and for u = 105 (Fig. 3d) the thermal irreversibilities predominate, because Bejan number is above 1/2. Besides, a good concordance in results is observed.
The present numerical implementation was validated by reproducing the results of two authors. The results of entropy generation are compared with those presented by Magherbi et al. [20] and the results of Nusselt number with that of the Val Davis and Jones [26]. It should be noted that it was used the same Prandtl numbers that were used in this different works (Pr = 0.7 to compare the entropy generation and Pr = 1.0 to compare the Nusselt number). 3.1.1. Entropy generation Although all the cases presented by Magherbi et al. [20] have been simulated, with good concordance among the results, just some comparisons are presented in this work. The first comparison is with the variation of dimensionless total entropy generation (S_ T;a ) with time, showed in Fig. 2a and b. In Fig. 2a the transient entropy generation is showed for the case of Ra = 103, with irreversibility distribution u = 102, while in Fig. 2b is showed the results for the case of Ra = 105, with irreversibility distribution of 104. Comparing the two figures it can be observed that both the order of magnitude of entropy genera-
Fig. 3. Bejan number vs. dimensionless time to Ra = 104: (a) u = 102, (b) u = 103, (c) u = 104, and (d) u = 105.
Fig. 2. Total entropy generation vs. dimensionless time. (a) Ra = 103, u = 102 and (b) Ra = 105, u = 104.
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3.1.2. Nusselt number To do the numerical validation with the Nusselt number it was decided to reproduce the results presented by Val Davis and Jones [26]. These authors presented the average Nusselt number of the hot wall. In Table 1 is presented the numerical validation through Nusselt number, comparing the results of the present work with those presented by Val Davis and Jones [26]. In this table, it can be observed a good concordance between the results. 3.2. Case study As previously mentioned, one of the objectives of this work is to observe the variation of the total entropy generation in relation to Rayleigh number and to the aspect ratio in rectangular cavities. To do this, the natural convection process in rectangular cavities was simulated with different aspect ratios (A = 0.25, 0.5, 1, 2, and 4). For
Table 1 Nusselt number vs. Rayleigh number Nu Ra = 104
Ra = 105
Ra = 106
1.118 1.116
2.243 2.239
4.519 4.531
8.800 8.726
a
b
107 ϕ = 10
107
−5
106
ϕ = 10−4
106
A=4
A=4
A=2
105
105
A=2
A=1
A=1
A=0.5
104
104
A=0.5
2
S T, p
10
A=0.25
10
3
10
2
.
10
.
S T, p
A=0.25 3
101 10 10
101
0
10
-1
10 102
103
104
105
106
107
0
-1
108
102
103
104
Ra
c
10
d
ϕ = 10−3 106
10
3
10
2
107
108
10
106
107
108
ϕ=10−2
A=2
A=4 A=2
105
A=1 A=0.5
S T, p
A=0.25
10
4
10
3
10
2
A=1 A=0.5 A=0.25
.
S T, p
10
4
106
7
106
A=4
105
105
Ra
7
.
Ref. [26] Present
Ra = 103
each aspect ratio, the process was simulated with different Rayleigh number values (Ra = 103, 104, 105, 106, 107), and, for each Rayleigh number, four irreversibility coefficients (u = 102, 103, 104, 105), that totalized 100 different cases. The mathematical model that was used is the same presented before, in this case with Pr = 0.7. With the objective to achieve spatial grid independence, the cases that presented higher velocity and temperature gradients (Ra = 107) were simulated with different convergence criteria values and several spatial and temporal grids. The convergence criterion is the mass residue and it was fixed in 108. The spatial grids were generated from the volume distribution equation, which was presented by Davidson [25]. For Ra = 105, grids of 30 30, 40 40, 60 60, 70 70, 80 80, and 90 90 volumes were tested, with three refinement coefficients (1.5, 2.0, and 2.5). In these conditions, it was observed that with a refinement coefficient equal to 2.0 and with grids of 50 50 volumes the values of total entropy generation showed differences lower than 1% between grids. In order to guarantee a good numerical precision, it was decided to simulate all the cases with grid of 70 70 volumes and refinement coefficient equal 2.0. The dependence of the total entropy generation in function of the aspect ratio, of the irreversibility coefficient and of the Rayleigh number, in steady state, is shown graphically in Fig. 4a–d. In these figures, it can be observed that the total entropy generation
101 10 10
101
0
10
-1
10 102
103
104
105
Ra
106
107
108
0
-1
102
103
104
105
Ra
Fig. 4. Variation of total entropy generation vs. Rayleigh number to A = 0.25, 0.5, 1, 2, and 4: (a) u = 105, (b) u = 104, (c) u = 103, and (d) u = 102.
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increases in function of the aspect ratio and also increases exponentially as function of the Rayleigh number. Also it can be observed that the total entropy generation is much influenced by the irreversibility coefficient. Besides, there is proportionality between the magnitude order of the variation of the entropy generation and the irreversibility coefficients. Fig. 5a–d shows the influence of the aspect ratio, irreversibility coefficient and Rayleigh number on Bejan number. In Fig. 5a (u = 105) it can be observed that, for Ra = 103 and 104, independently of the aspect ratio, the entropy generation is dominated by the thermal effects, since, in this case, the Bejan number decreases with the increase of the Rayleigh number and of the aspect ratio. The higher the Rayleigh number and smaller the irreversibility coefficient the higher the influence of the aspect ratio on Bejan number. As the irreversibility coefficient increases the viscous effects starts to compete with the thermal effects, as it can be observed in Fig. 5b–d through Bejan number. Be ffi 1 in the Fig. 5a (u = 103) shows that, for Ra = 103 and Ra = 104, the irreversibilities due to the thermal effects are predominant. In Fig. 5b (u = 104) this happens only for Ra = 103. In Fig. 5c–d (u = 103 and u = 102) it can be observed that, despite the thermal irreversibilities dominating the process, for Rayleigh numbers up to the order of 105, the viscous irreversibilities are also present,
a
and their influence on entropy generation is also associated to the aspect ratio. The variation of entropy generation due to viscous effects for five aspects ratios and four irreversibility coefficients versus Rayleigh number can be observed in Fig. 6a–d. In these figures, it can be seen that the entropy generation due to viscous effects increases exponentially as Rayleigh number increases. It also can be observed that the smaller the Rayleigh numbers the bigger the influence of the aspect ratio on entropy generation. This characteristic is associated to the influence of the hydrodynamic boundary layer near the vertical walls and to the entropy generation itself. In this case, the influence of entropy generation is associated to the viscous effects (Eq. (13)) that considers the velocity gradients in its definition. In natural convection process the biggest velocity gradients are found near the walls which have bigger thermal gradients. This happens because in these places the fluid is accelerated due to the action of the buoyant force, as an exigency of mass conservation. Therefore, for the same Rayleigh number, the walls that have bigger dimensions generate bigger entropy due to the viscous effects than those walls that present smaller dimensions. Fig. 7a–c show these characteristics for Ra = 103 and u = 102. To facilitate the comparison these figures were kept in the same scale. In this case, the visualization of the process in cavities that have A = 0.5
b
1
10
0
10-1
0.1
Be
ϕ = 10−5
Be
A=4 A=2
ϕ = 10−4
10-2
A=4
A=1
0.01
A=2
A=0.5
A=1
A=0.25
10
-3
A=0.5 A=0.25
10-4
0.001 102
103
104
105
106
107
102
108
103
104
10
d
0
107
108
106
107
108
100
10-1
10-1
10
106
Ra
Ra
c
105
10
-2
10
-3
10
-4
-2
Be
Be
ϕ = 10−3 A=4
10-3
ϕ = 10−2
A=2 A=1
A=4 A=2
A=0.5
10-4
A=1
A=0.25
10-5
A=0.5 A=0.25
10
-5
10-6 102
103
104
105
Ra
106
107
108
102
103
104
105
Ra
Fig. 5. Bejan number vs. Rayleigh number to A = 0.25, 0.5, 1, 2 and 4: (a) u = 105, (b) u = 104, (c) u = 103, and (d) u = 102.
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a
b
108 107
107
A=4
10
105
A=2
105
4
A=1
10
4
A=0.5
103
10
10
ϕ = 10
6
A=0.25
10
1
10
0
f, p
102
.S
f, p
103
.S
108
−5
10
1
10
0
10
10
10
-2
10
-2
10
-3
10
-3
10
-4
10
-4
10
-5
10
-5
10
10
10
4
10
5
10
6
10
7
10
A=2 A=1 A=0.5 A=0.25
102
-1
3
A=4
6
-1
2
ϕ = 10−4
8
102
103
104
Ra 108
d
107
10
6
10
10
5
105
10
4
104
10
3
103
10
2
10
1
10
0
ϕ = 10−3 A=4 A=2
10-1 10 10
-3
10
-4
10
-5
f, p
10
-2
A=1 A=0.5 A=0.25
102
103
104
105
106
107
108
108
7
.S
.S
f, p
c
105
Ra
106
107
108
6
102 ϕ = 10−2
10
1
10
0
A=4
10-1
A=2
10
-2
A=1
10
-3
10
-4
10
-5
A=0.5 A=0.25
102
103
Ra
104
105
106
107
108
Ra
Fig. 6. Variation of entropy generation due to viscous effects vs. Rayleigh number to A = 0.25, 0.5, 1, 2 and 4: (a) u = 105, (b) u = 104, (c) u = 103, and (d) u = 102.
and A = 0.25 is not clear. For this reason, only three aspect ratios are presented: A = 4, 2, and 1. Fig. 7a shows the presence of the hydrodynamic boundary layer in all the extension of the two vertical walls. Besides this, in the same figure, the presence of a shear layer that separates the ascendant from the descendant hydrodynamic boundary layers can be observed in the center of the cavity. This figure shows that for this aspect ratio (A = 4) the velocity gradients are present in all analyzed system. Though the identification of the several velocity vectors present inside the boundary layer, this figure also indicates that the computational grid was sufficiently refined. Comparing Fig. 7a–c, it can be observed that, as the aspect ratio decreases, the hydrodynamic boundary layers draw away and show a region in the center of the cavity with disregarding velocity gradients. The increase of the Rayleigh number for the same aspect ratio also presents a separation between the two hydrodynamic boundary layers, as it can be seen in Figs. 8 and 9a–c, for u = 0.01 and 0.00001, respectively. In these cases, the separation becomes evident by reducing the hydrodynamic boundary layer thickness. This reduction is followed by an increase of the fluid acceleration, due to the buoyant force action. Comparing these two figures, it is observed that the flow keeps the qualitative characteristics for the same Rayleigh number. Therefore, if the thermal and geometric conditions of the flow are kept, the hydrodynamic characteristics are qualitatively similar and independent of the irreversibility
coefficient. The latter just interfere in the quantitative results as previously commented in relation to Figs. 4–6.
4. Conclusion This work presented an analysis of the entropy generation in rectangular cavities, submitted to the natural convection process due to the temperature difference between the vertical walls. In these processes, the amount of dislocated fluid is associated with the heat transfer rate and affects directly the entropy generated. For Bejan number greater than 0.5, the entropy generation is caused, mainly, by the heat transfer process. For Bejan number less than 0.5 the shearing forces predominate. The results of this work indicate that: (a) the total entropy generation in steady state increases linearly in both cases, the aspect ratio and the irreversibility coefficient, and exponentially with the Rayleigh number; (b) the influence of the aspect ratio on Bejan number is proportional to Rayleigh number and inversely proportional to the irreversibility coefficient; (c) for the same aspect ratio, the entropy generation due to the viscous effects increases with the Rayleigh number and, for a certain Rayleigh number, the entropy generation due to the viscous effects also increases with the aspect ratio. Finally, it was observed that, for a determined Rayleigh number, the entropy generation due to the viscous effects also increases according to the aspect ratio.
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ϕ=10-5
(a) A=4
Ra = 103 ϕ = 10-2 (b) A=2
(c) A=1
Fig. 7. Velocity fields to Ra = 103 and u = 102: (a) A = 4, (b) A = 2, and (c) A = 1.
(a) Ra=103
(b) Ra=105
(c) Ra=107
Fig. 9. Velocity fields to A = 4 and u = 105: (a) Ra = 103, (b) Ra = 105, and (c) Ra = 107.
ϕ=10-2
In this study it was possible to observe that the thermal and hydrodynamic problem is highly coupled. For a thermophysical configuration involving natural convection, the geometric configurations with minor aspect ratios are the better option. Decreasing the aspect ratio increase the systems efficiency. This type of geometrical and thermophysical configuration could be applied, for example, to the space between the glass cover and the absorber plate of a solar thermal collector, or, to the existing space between plates of electronic circuit, found in many equipments. Acknowledgement The authors thank the financial support from UNISINOS for the realization of this work. References
(a) Ra=103
(b) Ra=105
(c) Ra=107
Fig. 8. Velocity fields to A = 4 and u = 102: (a) Ra = 103, (b) Ra = 105, and (c) Ra = 107.
The usefulness of the entropy generation analysis in natural convection problem allows to get an indication of the possible geometry modifications aimed to enhance the thermal performance of the system by means of local entropy production.
[1] S. Mahmud, A.K.S. Island, Laminar free convection and entropy generation inside an inclined wavy enclose, Int. J. Therm. Sci. 42 (8) (2003) 1003– 1012. [2] A. Bejan, Heat Transfer, Wiley, New York, 1993. [3] J.E.B. Nelson, A.R. Balakrishnan, S.S. Murthy, Experiments on stratified chilledwater tanks, Int. J. Refrig. 22 (3) (1999) 216–234. [4] F. Ampofo, T.G. Karayiannis, Experimental benchmark data for turbulent natural convection in an air filled square cavity, Int. J. Heat Mass Transfer 46 (19) (2003) 3551–3572. [5] O.B. Adeyinka, G.F. Naterer, Experimental uncertainty of measured entropy production with pulsed laser PIV and planar laser induced fluorescence, Int. J. Heat Mass Transfer 48 (2005) 1450–1461. [6] F.T. Poujol, Natural convection of a high Prandtl number fluid in cavity, Int. Comm. Heat Mass Transfer 27 (1) (2000) 109–118. [7] R. De C. Oliveski, A. Krenzinger, H.A. Vielmo, Cooling of cylindrical vertical tank submitted to natural internal convection, Int. J. Heat Mass Transfer 46 (11) (2003) 2015–2026. [8] I. Zahmatkesh, On the importance of thermal boundary conditions in heat transfer and entropy generation for natural convection inside a porous enclosure, Int. J. Therm. Sci. 47 (3) (2008) 339–346.
R. De C. Oliveski et al. / Applied Thermal Engineering 29 (2009) 1417–1425 [9] A. Andreozzi, A. Auletta, O. Manca, Entropy generation in natural convection in a symmetrically and uniformly heated vertical channel, Int. J. Heat Mass Transfer 49 (2006) 3221–3228. [10] I. Dagtekin, H.F. Oztop, A. Bahloul, Entropy generation for natural convection in C-shaped enclosures, Int. Comm. Heat Mass Transfer 34 (2007) 502–510. [11] O.M. Haddad, M.K. Alkam, M.T. Khasawneh, Entropy generation due to laminar forced convection in the entrance region of a concentric annulus, Energy 29 (2004) 35–55. [12] A. Bejan, Second-law analysis in heat transfer and thermal design, Adv. Heat Transfer 15 (1982) 1–58. [13] A. Bejan, Entropy Generation Minimization, CRC Press, Boca Raton, New York, 1996. [14] H. Klalkhali, A. Faghri, Z.J. Zuo, Entropy generation in a pipe system, Appl. Therm. Eng. 19 (10) (1999) 1027–1043. [15] A.Z. Sahin, Entropy generation in turbulent liquid flow through a smooth duct subjected to constant wall temperature, Int. J. Heat Mass Transfer 43 (8) (2000) 1469–1478. [16] A. Sahin, Entropy generation and pumping power in a turbulent fluid flow through a smooth pipe subjected to constant heat flux, Exergy 2 (4) (2002) 314–321. [17] H. Abbassi, M. Magherbi, A.B. Brahim, Entropy generation in Poiseuille-Benard channel flow, Int. J. Therm. Sci. 42 (13) (2003) 1081–1088.
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[18] B.A.K. Abu-Hijleh, M. Abu-Qudais, E. Abu-Nada, Numerical prediction of entropy generation due to natural convection from a horizontal cylinder, Energy 24 (4) (1999) 327–333. [19] B.A.K. Abu-Hijleh, W.N. Heilen, Entropy generation due to laminar natural convection over a heated rotating cylinder, Int. J. Heat Mass Transfer 42 (22) (1999) 4225–4233. [20] M. Magherbi, H. Abbassi, A.B. Brahim, Entropy generation at the onset of natural convection, Int. J. Heat Mass Transfer 46 (18) (2003) 3441–3450. [21] S. Mahmud, R.A. Fraser, The second law analysis in fundamental convective heat transfer problems, Int. J. Therm. Sci. 42 (2) (2003) 177–186. [22] A. Bejan, Entropy Generation Through Heat and Fluid Flow, Wiley, New York, 1982. [23] S.V. Patankar, Numerical Heat Transfer and Fluid Flow, McGraw-Hill, Washington, 1980. [24] J.P. Van Doormaal, G.D. Raithby, Enhancements of the simple method for predicting incompressible fluid flow, Numer. Heat Transfer 7 (1984) 147–163. [25] L. Davidson, Calculation of the turbulence buoyancy-driven flow in a rectangular cavity using an efficient solver and two different low Reynolds number j–e turbulence models, Numer. Heat Transfer (A) 18 (1990) 129–147. [26] G.D. Val Davis, I.P. Jones, Natural convection in a square cavity: a comparison exercise, Int. J. Numer. Methods Fluids 3 (1993) 227–248.
Contents lists available at ScienceDirect
Applied Thermal Engineering journal homepage: www.elsevier.com/locate/apthermeng
Review
Entropy generation and natural convection in rectangular cavities Rejane De C. Oliveski *, Mario H. Macagnan, Jacqueline B. Copetti Department of Mechanical Engineering, School of Physical and Technological Sciences, Universidade do Vale do Rio dos Sinos, UNISINOS, Av. Unisinos 950 São Leopoldo, CEP 93020-000, RS, Brazil
a r t i c l e
i n f o
Article history: Received 14 September 2006 Accepted 15 July 2008 Available online 23 July 2008 Keywords: Entropy Natural convection Bejan number Rayleigh number
a b s t r a c t This work presents a numerical analysis of entropy generation in rectangular cavities that were submitted to the natural convection process. This natural convection process was caused by temperature differences between the vertical walls of the cavities. Momentum and energy equations were used to solve this problem. These equations were coupled by the Boussinesq approximation. Initially the cavities were submitted to uniform temperature and velocity fields. The hypothesis of perfect insulation was considered for the top and bottom walls of the cavity. Impermeability and non-slip condition in the boundary were assumed for every wall of the cavity. The numerical analysis is performed through a two-dimensional model with the Finite Volume method. The results of the entropy generation obtained to a square cavity were used to validate the numerical model and it presented good concordance with results from other authors. Additionally, an analysis of the entropy generation in rectangular cavities was performed with five aspect ratios, five Rayleigh numbers and four irreversibility coefficients. The results of this work indicate that: (a) the total entropy generation in steady state increases linearly in both cases, the aspect ratio and the irreversibility coefficient, and exponentially with the Rayleigh number; (b) the influence of the aspect ratio on Bejan number is proportional to Rayleigh number and inversely proportional to the irreversibility coefficient; (c) for the same aspect ratio, the entropy generation due to the viscous effects increases with the Rayleigh number and, for a certain Rayleigh number, the entropy generation due to the viscous effects also increases with the aspect ratio. Ó 2008 Elsevier Ltd. All rights reserved.
Contents 1. 2.
3.
4.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mathematical model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. Initial and boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. Entropy generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3. Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Numerical validation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1. Entropy generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.2. Nusselt number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Case study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1. Introduction Natural convection heat transfer is present in some important processes in engineering such as thermal storage, environmental * Corresponding author. Tel.: +55 51 3591 1100x1719. E-mail address: [email protected] (R. De C. Oliveski). 1359-4311/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.applthermaleng.2008.07.012
1417 1418 1419 1419 1420 1420 1420 1420 1421 1421 1423 1424 1424
comfort, grain drying, electronic cooling and others. Whatever the shape of the wall, flow and heat transfer problems inside enclosures have numerous engineering applications like solar-collectors, double-wall insulation, electric machinery, cooling system of electronic devices, natural circulation in the atmosphere, etc. [1]. In natural convection processes, the thermal and the hydrodynamic are coupled and both are, according to Bejan [2], strongly
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Nomenclature A Be Cp g h H k L Nu p P Pr Ra S_ t T u, v U, V x, y X, Y
aspect ratio, H/L Bejan number, S_ l;a;h =S_ l;a specific heat at constant pressure (J/kg K) gravitational acceleration (m/s2) heat transfer coefficient (W/m2 K) cavity height (m) thermal conductivity (W/m K) cavity length (m) Nusselt number, hL/k pressure (Pa) dimensionless pressure, pL2/qa2 Prandtl number, t/a Rayleigh number, (gbDTL3)/(at) entropy generation (W/m3 K) time (s) temperature (°C) velocity components in x, y direction (m/s) dimensionless velocity, components in x, y direction, uL/a, mL/a cartesian coordinates (m) dimensionless cartesian coordinates, x/L, y/L
influenced by the fluid thermophysical characteristics, the temperature differences and the system geometry. The existence of a large number of systems submitted to the natural convection process, involving different physical and constructive characteristics, makes the study of this phenomenon an always present theme, despite the several decades of research on the subject. Several works about natural convection in many different areas are available, with experimental [3–5] and numerical approaches [6–11]. All real processes present irreversibilities and can be associated to friction, mass transference, thermal gradients, chemical reactions, etc., resulting in process efficiency loss. This efficiency loss is related to the entropy generation. As the generation of entropy destroys the system energy, its minimization has been used as the optimal design criteria for thermal systems. The mechanisms of entropy generation shall be understood [12,13] and such understanding can be achieved with a detailed study of the process, with the analysis of this dependence with the different thermal, hydrodynamic and geometric characteristics of these processes. In this way, an optimal configuration with minimum loss of available energy may be obtained. Many entropy generation works about flows with heat transfer, in both case, natural and forced convection regime, can be found in the literature. In the case of forced convection regime it can be mentioned the works of Haddad et al. [11], Khalkhali et al. [14], Sahin [15,16] and Abbassi et al. [17], among others. Concerning entropy generation in natural convection regime it can be mentioned the works of Zahmatkesh [8], Abu-Hijleh et al. [18], Abu-Hijleh and Heilen [19], Mahmud and Island [1], Magherbi et al. [20] and others. Sahin [15] analyzed the variation of entropy generation in function of viscosity in forced flow. For this problem, the authors prescribed a constant temperature to the duct walls. In another work, the same author [16] made a similar analysis with constant heat flow imposed to the duct wall. Mahmud and Fraser [21] developed analytical solutions for entropy generation, irreversibility distribution and Bejan number for a series of classical forced convection problems, from their analytic solution of velocity and temperature profile. For a range of Rayleigh numbers and irrever-
Greeks
a b
l m q f h u
thermal diffusivity (m2/s) thermal expansion coefficient (K1) dynamic viscosity (kg/ms) kinematic viscosity (m2/s) density (kg/m3) dimensionless time, ta/L2 dimensionless temperature, (T Tc)/(Th Tc) irreversibility ratio, (lT0/k)(a/L(Th Tc))2
Subscripts a dimensionless c could f fluid viscous effect h hot, heat transfer l local max maximum p permanent T total
sibilities, Abbassi et al. [17] investigated numerically the entropy generation in Poiseuille–Bernard flows, where Reynolds and Peclet numbers were kept constant. The entropy generation due to natural convection in a horizontal cylinder was investigated numerically by Abu-Hijleh et al. [18]. In this work, the authors showed that, for a range of Rayleigh number, the entropy generation decreases according to the increase of the cylinder diameter. They also showed that the inverse happens when the Rayleigh number is increased in relation to the same diameter. Recently, Zahmatkesh [8] analyzed the importance of thermal boundary conditions in heat transfer process and entropy generation characteristics inside of a porous enclosure. To do this, the author simulated the natural convection processes in a porous enclosure over a wide range of Darcy-modified Rayleigh number. Zahmatkesh [8] concluded that the generation rate is likely to be the highest for uniform heating/cooling and the lowest for non-uniform heating and that the optimum case with respect to heat transfer and entropy generation could be achieved by non-uniform heating. The aim of the present work is the study of entropy generation in natural convection processes in rectangular cavity through numerical simulation. The cavity was submitted to a horizontal temperature gradient, inducing the natural convection process inside the cavity. These conditions could be applied for solar thermal collectors and reservoirs, cooling in electronic enclosures and others. Initially, a numerical validation is presented, comparing some results of this work with that of entropy generation found in the literature for a square cavity. The contribution of this work is the analyses of the variation of entropy generation in relation to Rayleigh number, aspect ratio and irreversibility coefficient in rectangular cavities submitted to natural convection process, allowing identifying the source and location of entropy generation. 2. Mathematical model To solve the entropy generation problem due to natural convection in rectangular cavities, it is assumed that: the fluid is incompressible, the Boussinesq approximation is valid and that the flow has two-dimensional characteristics, as indicated in Fig. 1.
R. De C. Oliveski et al. / Applied Thermal Engineering 29 (2009) 1417–1425
y
oV o o oP o oV þ ðUVÞ þ ðVVÞ ¼ þ Pr of oX oY oY oX oX o oV þ Pr þ Pr Rah; oY oY oh o o o oh o oh þ ; þ ðUhÞ þ ðVhÞ ¼ of oX oY oX oX oY oY
∂T ∂y = 0
Th
Tc
g
u x
L
2.2. Entropy generation
Fig. 1. Schematic view of cavity.
Furthermore, it is assumed that the fluid in the cavities is Newtonian and has constant properties. In this case, the governing equations are the ones that follow:
o o ð1Þ ðquÞ þ ðqvÞ ¼ 0; ox oy o o o op o ou o ou l l ðquÞ þ ðquuÞ þ ðqvuÞ ¼ þ þ ; ot ox oy ox ox ox oy oy
ð3Þ ð4Þ
where u and v are, respectively, the velocities in ‘‘x” and ‘‘y” directions, q is the density, p is the pressure, l is the absolute viscosity, and b is the fluid thermal expansion coefficient. The variable a represents the thermal diffusivity, T, the temperature, t, time, and g, gravity acceleration. The dimensionless form of the Eqs. (1)–(4) can be obtained by applying the following parameters:
h¼
at
T Tc ; Th Tc
;
Pr ¼
m ta H ; f¼ 2; A¼ ; L a L
ðX; YÞ ¼
x y ; ; L L
ðU; VÞ ¼
uL vL ; ;
a a
In the natural convection process the entropy generation is associated to the heat transfer and to the fluid flow friction. According to Bejan [22], the local entropy generation (S_ l ) can be determined by the following expression:
2 2 # oT oT þ ox oy " 2 2 # 2 l ou ov ou ov þ ; 2 þ2 þ þ ox oy oy ox T0
k S_ l ¼ 2 T0
ð2Þ
o o o op o ov l ðqvÞ þ ðquvÞ þ ðqvvÞ ¼ þ ot ox oy oy ox ox o ov þ l þ qbgðT T C Þ; oy oy oT o o o oT o oT a a þ ðuTÞ þ ðvTÞ ¼ þ ; ot ox oy ox ox oy oy
gbDTL3
ð9Þ
The boundary conditions for the hydrodynamic problem are impermeability and non-slipping in all cavity walls. For the thermal problem, the horizontal walls were kept adiabatic while the vertical walls were kept at a constant temperature. The hot wall was kept with h = 0.5 and cold wall with h = 0.5. In all domain the initial conditions were: zero to the velocity field and h = 0.5 X to the temperature field.
v
Ra ¼
ð8Þ
2.1. Initial and boundary conditions
H
∂T ∂y = 0
1419
P¼
pL2
qa
2
;
where Ra and Pr are Rayleigh and Prandtl numbers, respectively, v is the kinematic viscosity, Tc is the cold wall temperature, Th is the hot wall temperature and h is the dimensionless temperature. The time is represented by t and the dimensionless time by f, the pressure by p and the dimensionless pressure by P. Besides these variables, A is the aspect cavity ratio, U and V are the dimensionless velocities in X and Y directions, respectively, also dimensionless. After the replacement of this parameters in Eqs. (1)–(4) it can be obtained the dimensionless equations of the problem (Eqs. (6)– (9)), as presented in [20].
ð6Þ ð7Þ
ð10Þ
where k is the thermal conductivity of the fluid. The first term of Eq. (10) represents the dimensional entropy generation due to heat transfer (S_ l;h ) while the second term represents the entropy generation due to the viscous effects of the fluid (S_ l;f ), so that
S_ ¼ S_ l;h þ S_ l;f :
ð11Þ
This equation can be rewritten by using the dimensionless parameters showed in Eq. (5), that results in the following expressions:
2 2 # oh oh ; þ oX oY " 2 2 # 2 oU oV oU oV ; ¼u 2 þ2 þ þ oX oY oY oX
S_ l;a;h ¼ S_ l;a;f
"
S_ l;a ¼ S_ l;a;h þ S_ l;a;f ; 2 lT a u¼ 0 ; k LðT h T c Þ ð5Þ
oU oV þ ¼ 0; oX oY oU oðUU Þ oðVV Þ oP o oU o oU þ þ ¼ þ Pr þ Pr ; of oX oY oX oX oX oY oY
"
ð12Þ ð13Þ ð14Þ
Th þ Tc T0 ¼ ; 2
ð15Þ
where T0 is the bulk temperature and u is the ratio between the viscous and thermal irreversibilities. The dimensionless total entropy generation (S_ t;a ) is obtained through the integration of the local entropy generation (S_ l;a ) in all computational domain, as indicated in Eq. (16).
S_ T; a ¼
Z
S_ l;a dv:
ð16Þ
v
An alternative parameter for irreversibilities distribution is the Bejan number (Be) defined as [20]
Be ¼
S_ l;a;h : S_ l;a
ð17Þ
The heat transfer irreversibilities are dominant when Be 1/2. When Be 1/2 the irreversibilities due to the viscous effects dominate the processes and if Be = 1/2 the entropy generation due to the viscous effects and the heat transfer are equal [20].
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2.3. Methodology The numerical solution is obtained by the discretization of Eqs. (6)–(9), in Finite Volume method, as described by Pantakar [23]. The pressure–velocity coupling is obtained by the SIMPLEC method from Van Doormaal and Raithby [24]. The Power Law scheme is used for interpolation in the faces of the control volumes. The resulting equation systems are solved by the TDMA algorithm. The convergence criterion used was the mass conservation residue, established in 108. It was used a spatial grid non-uniformly spaced, with 70 70 volume elements, with volume concentration in the cavities walls. For that, an equation for volume distribution suggested by Davidson [25] it was used, with a grid refinement coefficient equal to 2.0. 3. Results Two types of analysis are presented in this work. The first on represents the validation of the mathematical model and its implementation (Section 3.1). The second shows the contribution of this work in relation of the entropy generation in natural convection process (Section 3.2). 3.1. Numerical validation
tion and the curve behavior are different. In this figure, it can be also observed that, after the transient period, the entropy generation remains practically constant. On the other hand, in Fig. 2b, it can be observed that after a high increase of entropy generation, it decreases significantly, with further oscillations before reaching the steady state regime. There are some explanations in the literature [19,20] for this fact. Magherbi et al. [20] says that for small Rayleigh numbers the steady state is very close to the equilibrium state and therefore the system reaches the steady state regime rapidly, without passing through the oscillation period. On the other hand, for Ra > Rac = 5200 the steady state regime draws away from the equilibrium state. In this case, oscillations are observed around the stationary state, which correspond to the total entropy generation oscillations. By definition, if Bejan number is equal to 1/2 it means that the thermal and viscous irreversibilities are equivalent. Above this value the thermal irreversibilities predominate. In Fig. 3, Bejan number is presented in function of the dimensionless time, which was obtained with Eq. (17) for Ra = 104 and four values of irreversibility distribution. These results are compared to that one that was obtained by Magherbi et al. [20]. In Fig. 3, it can be observed that for u = 104 (Fig. 3c) and for u = 105 (Fig. 3d) the thermal irreversibilities predominate, because Bejan number is above 1/2. Besides, a good concordance in results is observed.
The present numerical implementation was validated by reproducing the results of two authors. The results of entropy generation are compared with those presented by Magherbi et al. [20] and the results of Nusselt number with that of the Val Davis and Jones [26]. It should be noted that it was used the same Prandtl numbers that were used in this different works (Pr = 0.7 to compare the entropy generation and Pr = 1.0 to compare the Nusselt number). 3.1.1. Entropy generation Although all the cases presented by Magherbi et al. [20] have been simulated, with good concordance among the results, just some comparisons are presented in this work. The first comparison is with the variation of dimensionless total entropy generation (S_ T;a ) with time, showed in Fig. 2a and b. In Fig. 2a the transient entropy generation is showed for the case of Ra = 103, with irreversibility distribution u = 102, while in Fig. 2b is showed the results for the case of Ra = 105, with irreversibility distribution of 104. Comparing the two figures it can be observed that both the order of magnitude of entropy genera-
Fig. 3. Bejan number vs. dimensionless time to Ra = 104: (a) u = 102, (b) u = 103, (c) u = 104, and (d) u = 105.
Fig. 2. Total entropy generation vs. dimensionless time. (a) Ra = 103, u = 102 and (b) Ra = 105, u = 104.
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3.1.2. Nusselt number To do the numerical validation with the Nusselt number it was decided to reproduce the results presented by Val Davis and Jones [26]. These authors presented the average Nusselt number of the hot wall. In Table 1 is presented the numerical validation through Nusselt number, comparing the results of the present work with those presented by Val Davis and Jones [26]. In this table, it can be observed a good concordance between the results. 3.2. Case study As previously mentioned, one of the objectives of this work is to observe the variation of the total entropy generation in relation to Rayleigh number and to the aspect ratio in rectangular cavities. To do this, the natural convection process in rectangular cavities was simulated with different aspect ratios (A = 0.25, 0.5, 1, 2, and 4). For
Table 1 Nusselt number vs. Rayleigh number Nu Ra = 104
Ra = 105
Ra = 106
1.118 1.116
2.243 2.239
4.519 4.531
8.800 8.726
a
b
107 ϕ = 10
107
−5
106
ϕ = 10−4
106
A=4
A=4
A=2
105
105
A=2
A=1
A=1
A=0.5
104
104
A=0.5
2
S T, p
10
A=0.25
10
3
10
2
.
10
.
S T, p
A=0.25 3
101 10 10
101
0
10
-1
10 102
103
104
105
106
107
0
-1
108
102
103
104
Ra
c
10
d
ϕ = 10−3 106
10
3
10
2
107
108
10
106
107
108
ϕ=10−2
A=2
A=4 A=2
105
A=1 A=0.5
S T, p
A=0.25
10
4
10
3
10
2
A=1 A=0.5 A=0.25
.
S T, p
10
4
106
7
106
A=4
105
105
Ra
7
.
Ref. [26] Present
Ra = 103
each aspect ratio, the process was simulated with different Rayleigh number values (Ra = 103, 104, 105, 106, 107), and, for each Rayleigh number, four irreversibility coefficients (u = 102, 103, 104, 105), that totalized 100 different cases. The mathematical model that was used is the same presented before, in this case with Pr = 0.7. With the objective to achieve spatial grid independence, the cases that presented higher velocity and temperature gradients (Ra = 107) were simulated with different convergence criteria values and several spatial and temporal grids. The convergence criterion is the mass residue and it was fixed in 108. The spatial grids were generated from the volume distribution equation, which was presented by Davidson [25]. For Ra = 105, grids of 30 30, 40 40, 60 60, 70 70, 80 80, and 90 90 volumes were tested, with three refinement coefficients (1.5, 2.0, and 2.5). In these conditions, it was observed that with a refinement coefficient equal to 2.0 and with grids of 50 50 volumes the values of total entropy generation showed differences lower than 1% between grids. In order to guarantee a good numerical precision, it was decided to simulate all the cases with grid of 70 70 volumes and refinement coefficient equal 2.0. The dependence of the total entropy generation in function of the aspect ratio, of the irreversibility coefficient and of the Rayleigh number, in steady state, is shown graphically in Fig. 4a–d. In these figures, it can be observed that the total entropy generation
101 10 10
101
0
10
-1
10 102
103
104
105
Ra
106
107
108
0
-1
102
103
104
105
Ra
Fig. 4. Variation of total entropy generation vs. Rayleigh number to A = 0.25, 0.5, 1, 2, and 4: (a) u = 105, (b) u = 104, (c) u = 103, and (d) u = 102.
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increases in function of the aspect ratio and also increases exponentially as function of the Rayleigh number. Also it can be observed that the total entropy generation is much influenced by the irreversibility coefficient. Besides, there is proportionality between the magnitude order of the variation of the entropy generation and the irreversibility coefficients. Fig. 5a–d shows the influence of the aspect ratio, irreversibility coefficient and Rayleigh number on Bejan number. In Fig. 5a (u = 105) it can be observed that, for Ra = 103 and 104, independently of the aspect ratio, the entropy generation is dominated by the thermal effects, since, in this case, the Bejan number decreases with the increase of the Rayleigh number and of the aspect ratio. The higher the Rayleigh number and smaller the irreversibility coefficient the higher the influence of the aspect ratio on Bejan number. As the irreversibility coefficient increases the viscous effects starts to compete with the thermal effects, as it can be observed in Fig. 5b–d through Bejan number. Be ffi 1 in the Fig. 5a (u = 103) shows that, for Ra = 103 and Ra = 104, the irreversibilities due to the thermal effects are predominant. In Fig. 5b (u = 104) this happens only for Ra = 103. In Fig. 5c–d (u = 103 and u = 102) it can be observed that, despite the thermal irreversibilities dominating the process, for Rayleigh numbers up to the order of 105, the viscous irreversibilities are also present,
a
and their influence on entropy generation is also associated to the aspect ratio. The variation of entropy generation due to viscous effects for five aspects ratios and four irreversibility coefficients versus Rayleigh number can be observed in Fig. 6a–d. In these figures, it can be seen that the entropy generation due to viscous effects increases exponentially as Rayleigh number increases. It also can be observed that the smaller the Rayleigh numbers the bigger the influence of the aspect ratio on entropy generation. This characteristic is associated to the influence of the hydrodynamic boundary layer near the vertical walls and to the entropy generation itself. In this case, the influence of entropy generation is associated to the viscous effects (Eq. (13)) that considers the velocity gradients in its definition. In natural convection process the biggest velocity gradients are found near the walls which have bigger thermal gradients. This happens because in these places the fluid is accelerated due to the action of the buoyant force, as an exigency of mass conservation. Therefore, for the same Rayleigh number, the walls that have bigger dimensions generate bigger entropy due to the viscous effects than those walls that present smaller dimensions. Fig. 7a–c show these characteristics for Ra = 103 and u = 102. To facilitate the comparison these figures were kept in the same scale. In this case, the visualization of the process in cavities that have A = 0.5
b
1
10
0
10-1
0.1
Be
ϕ = 10−5
Be
A=4 A=2
ϕ = 10−4
10-2
A=4
A=1
0.01
A=2
A=0.5
A=1
A=0.25
10
-3
A=0.5 A=0.25
10-4
0.001 102
103
104
105
106
107
102
108
103
104
10
d
0
107
108
106
107
108
100
10-1
10-1
10
106
Ra
Ra
c
105
10
-2
10
-3
10
-4
-2
Be
Be
ϕ = 10−3 A=4
10-3
ϕ = 10−2
A=2 A=1
A=4 A=2
A=0.5
10-4
A=1
A=0.25
10-5
A=0.5 A=0.25
10
-5
10-6 102
103
104
105
Ra
106
107
108
102
103
104
105
Ra
Fig. 5. Bejan number vs. Rayleigh number to A = 0.25, 0.5, 1, 2 and 4: (a) u = 105, (b) u = 104, (c) u = 103, and (d) u = 102.
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a
b
108 107
107
A=4
10
105
A=2
105
4
A=1
10
4
A=0.5
103
10
10
ϕ = 10
6
A=0.25
10
1
10
0
f, p
102
.S
f, p
103
.S
108
−5
10
1
10
0
10
10
10
-2
10
-2
10
-3
10
-3
10
-4
10
-4
10
-5
10
-5
10
10
10
4
10
5
10
6
10
7
10
A=2 A=1 A=0.5 A=0.25
102
-1
3
A=4
6
-1
2
ϕ = 10−4
8
102
103
104
Ra 108
d
107
10
6
10
10
5
105
10
4
104
10
3
103
10
2
10
1
10
0
ϕ = 10−3 A=4 A=2
10-1 10 10
-3
10
-4
10
-5
f, p
10
-2
A=1 A=0.5 A=0.25
102
103
104
105
106
107
108
108
7
.S
.S
f, p
c
105
Ra
106
107
108
6
102 ϕ = 10−2
10
1
10
0
A=4
10-1
A=2
10
-2
A=1
10
-3
10
-4
10
-5
A=0.5 A=0.25
102
103
Ra
104
105
106
107
108
Ra
Fig. 6. Variation of entropy generation due to viscous effects vs. Rayleigh number to A = 0.25, 0.5, 1, 2 and 4: (a) u = 105, (b) u = 104, (c) u = 103, and (d) u = 102.
and A = 0.25 is not clear. For this reason, only three aspect ratios are presented: A = 4, 2, and 1. Fig. 7a shows the presence of the hydrodynamic boundary layer in all the extension of the two vertical walls. Besides this, in the same figure, the presence of a shear layer that separates the ascendant from the descendant hydrodynamic boundary layers can be observed in the center of the cavity. This figure shows that for this aspect ratio (A = 4) the velocity gradients are present in all analyzed system. Though the identification of the several velocity vectors present inside the boundary layer, this figure also indicates that the computational grid was sufficiently refined. Comparing Fig. 7a–c, it can be observed that, as the aspect ratio decreases, the hydrodynamic boundary layers draw away and show a region in the center of the cavity with disregarding velocity gradients. The increase of the Rayleigh number for the same aspect ratio also presents a separation between the two hydrodynamic boundary layers, as it can be seen in Figs. 8 and 9a–c, for u = 0.01 and 0.00001, respectively. In these cases, the separation becomes evident by reducing the hydrodynamic boundary layer thickness. This reduction is followed by an increase of the fluid acceleration, due to the buoyant force action. Comparing these two figures, it is observed that the flow keeps the qualitative characteristics for the same Rayleigh number. Therefore, if the thermal and geometric conditions of the flow are kept, the hydrodynamic characteristics are qualitatively similar and independent of the irreversibility
coefficient. The latter just interfere in the quantitative results as previously commented in relation to Figs. 4–6.
4. Conclusion This work presented an analysis of the entropy generation in rectangular cavities, submitted to the natural convection process due to the temperature difference between the vertical walls. In these processes, the amount of dislocated fluid is associated with the heat transfer rate and affects directly the entropy generated. For Bejan number greater than 0.5, the entropy generation is caused, mainly, by the heat transfer process. For Bejan number less than 0.5 the shearing forces predominate. The results of this work indicate that: (a) the total entropy generation in steady state increases linearly in both cases, the aspect ratio and the irreversibility coefficient, and exponentially with the Rayleigh number; (b) the influence of the aspect ratio on Bejan number is proportional to Rayleigh number and inversely proportional to the irreversibility coefficient; (c) for the same aspect ratio, the entropy generation due to the viscous effects increases with the Rayleigh number and, for a certain Rayleigh number, the entropy generation due to the viscous effects also increases with the aspect ratio. Finally, it was observed that, for a determined Rayleigh number, the entropy generation due to the viscous effects also increases according to the aspect ratio.
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ϕ=10-5
(a) A=4
Ra = 103 ϕ = 10-2 (b) A=2
(c) A=1
Fig. 7. Velocity fields to Ra = 103 and u = 102: (a) A = 4, (b) A = 2, and (c) A = 1.
(a) Ra=103
(b) Ra=105
(c) Ra=107
Fig. 9. Velocity fields to A = 4 and u = 105: (a) Ra = 103, (b) Ra = 105, and (c) Ra = 107.
ϕ=10-2
In this study it was possible to observe that the thermal and hydrodynamic problem is highly coupled. For a thermophysical configuration involving natural convection, the geometric configurations with minor aspect ratios are the better option. Decreasing the aspect ratio increase the systems efficiency. This type of geometrical and thermophysical configuration could be applied, for example, to the space between the glass cover and the absorber plate of a solar thermal collector, or, to the existing space between plates of electronic circuit, found in many equipments. Acknowledgement The authors thank the financial support from UNISINOS for the realization of this work. References
(a) Ra=103
(b) Ra=105
(c) Ra=107
Fig. 8. Velocity fields to A = 4 and u = 102: (a) Ra = 103, (b) Ra = 105, and (c) Ra = 107.
The usefulness of the entropy generation analysis in natural convection problem allows to get an indication of the possible geometry modifications aimed to enhance the thermal performance of the system by means of local entropy production.
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