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Optimal fin geometry based on exergoeconomic analysis for a pin-fin array with application to electronics cooling
Exergy, an International Journal 2 (2002) 248–258 www.exergyonline.com
Optimal fin geometry based on exergoeconomic analysis for a pin-fin array with application to electronics cooling S.Z. Shuja Department of Mechanical Engineering, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia Received 6 May 2002; accepted 1 June 2002
Abstract Exergoeconomic analysis for a pin-fin array involves the achievement of a balance between the entropy generation due to heat transfer and pressure drop, while considering the unit cost of entropy generation. This process yields the optimum fin operation parameters based on minimum cost. In this study, analytical equations are presented considering the cost of operation for a pin-fin array. The solution of these equations would give the optimum fin diameter and length that result in a fin array with minimum operational cost. In addition, the influence of important fin thermal, physical, geometrical and cost parameters on the optimum diameter and length is presented in graphical form for quick calculations and easy interpretation. The presented results are subjected to the constraint that L/D is of the order of 1 or greater than 1. A case is also presented to demonstrate the use of the model for conditions typically found in cooling of electronic components. 2002 Éditions scientifiques et médicales Elsevier SAS. All rights reserved. Keywords: Thermoeconomic; Fin array; Optimization; Electronics cooling
1. Introduction Heat transfer enhancement devices are often employed to increase the rate of heat transfer. For applications in electronics cooling, the objective is to maintain the operating temperatures at a safe level for long term, reliable operation. In most cases, however, the selection of the most appropriate geometry for a particular application is difficult, since by adding a heat transfer enhancement device, not only the rate of heat transfer increases (reducing the heat transfer irreversibilities), but also the fluid friction increases (increasing the hydrodynamic irreversibilities). This raises the question as to what is the real advantage of the employed enhancement technique. To answer this question the well-known exergoeconomic analysis can be applied to the heat transfer enhancement devices. As described by Tsatsaronis [1], the exergoeconomic (thermoeconomic) analysis, evaluation, and optimization techniques are based on a combination of: (a) an exergy analysis, (b) a conventional economic analysis to calculate levelized costs, and (c) application of exergy costing, which recognizes that exergy provides the only rational basis for assigning costs to energy systems and energy carriers. The E-mail address: [email protected] (S.Z. Shuja).
exergoeconomic analysis identifies the cost associated with the real thermodynamic waste (exergy destruction and exergy loss) at the component level and reveals the real cost sources for each component. A comparison of the cost of exergy destruction within the component of a system with the component investment, operation, and maintenance costs for each component provides useful information for improving the cost effectiveness of the component as well as the overall system. An exergoeconomic analysis is very useful, particularly for complex thermal systems, in optimizing the overall system parameters or specific variables in a single component. A detailed description of the exergoeconomic analysis, evaluation, and optimization technique can be found in [2]. The continuous miniaturization of microelectronics, is posing new challenges in electronic cooling [3]. This initiated a wide research in the optimization of the heat-sink design. An overview of different types of heat sinks and associated design parameters are provided by Lee [4]. He also developed an analytical model for predicting and optimizing the thermal performance of bi-directional fin heat sinks in a partially confined configuration. Lau and Mahajan [3] performed experiments to examine the heat transfer and pressure drop characteristics for several heat sink geometries at various mass flow rates. They indicated that although the heat transfer increased as the fin density was increased, ac-
1164-0235/02/$ – see front matter 2002 Éditions scientifiques et médicales Elsevier SAS. All rights reserved. PII: S 1 1 6 4 - 0 2 3 5 ( 0 2 ) 0 0 0 8 1 - X
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Nomenclature a
fin array geometric parameter (1 for inline; 1/2 for staggered) A flow area defined in Eq. (A.3) . . . . . . . . . . . . . m2 B fluid friction irreversibility coefficient, Eq. (A.8) C, C1 , C2 , C3 parameters defined by Eq. (8) CD drag coefficient, Eq. (4) constant in Eq. (4) Cm Cn constant in Eq. (4) D diameter of the pin fin . . . . . . . . . . . . . . . . . . . . m F parameter defined by Eq. (A.8) G1 , G2 geometric parameters defined by Eq. (A.9) hf , hw , h heat transfer coefficient . . . . . . . . W·m−2 ·K−1 kf thermal conductivity of fluid . . . . . W·m−1 ·K−1 ks thermal conductivity of fin material . . . . . . . . . . . . . . . . . . . . . . . W·m−1 ·K−1 L length of the pin fin . . . . . . . . . . . . . . . . . . . . . . . m m constant in Eq. (4) m ˙ mass flow rate defined by Eq. (A.4) . . . . kg·s−1 mf fin constant defined by Eq. (A.3) M constant defined by Eq. (3) n constant in Eq. (4) N number of rows in a fin array Nu Nusselt number Pr Prandtl number P pressure drop . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pa qB base heat transfer . . . . . . . . . . . . . . . . . . . . . . . . W Reynolds number based on diameter ReD
companying this increased heat transfer was an increase in the pressure drop. Natural convection heat transfer in rectangular fin-arrays mounted on a vertical base was investigated experimentally by Guvenc and Yuncu [5]. Their results showed that there existed an optimum fin spacing for which the heat transfer rate from the fin array was maximized. Pierluigi and Enrico [6] performed numerical calculations for the local entropy generation in the flow around a heated finned tube. They determined optimal spacing of fins using entropy generation rate as well as total heat transfer as objective functions. Moores et al. [7] employed computational fluid dynamics and heat transfer techniques to model the thermal and hydrodynamic resistance characteristics through the pinfin structure of a prototype base plate design. They also verified the performance experimentally using water as the cooling fluid. Knight et al. [8] used an optimization scheme to design several air cooled aluminum finned arrays. They built and tested these designs experimentally and verified that the design obtained actually yield lowest thermal resistance, and hence, desirable operating temperatures, when used with the specified constraints. The optimal spacing and diameter for an isothermal fin array of fixed volume, appropriate for use with electronic equipment, was determined by Iyengar and
Reynolds number based on length entropy generation rate . . . . . . . . . . . . . . . W·K−1 entropy generation rate due to pressure drop, Eq. (A.10) . . . . . . . . . . . . . . . . . . . . . . . . . . W·K−1 ˙ ST entropy generation rate due to heat transfer, Eq. (A.11) . . . . . . . . . . . . . . . . . . . . . . . . . . W·K−1 Sn pin spacing in spanwise direction . . . . . . . . . . m Sp pin spacing in streamwise direction . . . . . . . . m T∞ absolute temperature of free stream . . . . . . . . . K maximum average fluid velocity occuring at the Umax minimum free flow area . . . . . . . . . . . . . . . m·s−1 V number of columns in the fin array W , X, Y , Z variables defined by Eq. (7) x variable defined by Eq. (12)
ReL S˙gen S˙P
Greek symbols θB ρ ν λH λP Γ˙ Γ˙ ∗
base stream temperature difference, = TB − T∞ density of fluid . . . . . . . . . . . . . . . . . . . . . . kg·m−3 kinematic viscosity of fluid . . . . . . . . . . . m2 ·s−1 unit cost of lost work due to heat transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . $·K·J−1 unit cost of lost work due to pressure drop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . $·K·J−1 cost of operation . . . . . . . . . . . . . . . . . . . . . . $·s−1 dimensionless cost of operation, = Γ˙ /C
Subscript opt
optimum
Bar-Cohen [9]. They discussed the influence of the pin-fin length on the optimal horizontal spacing and diameter, and the optimal heat dissipation with reference to a specified heat sink configuration. Morrison [10], presented the optimization of fin geometry for heat sinks in natural convection with rectangular cross-section fins at a constant fin spacing, and applied the method for steady state and intermittent duty cycle operation. Poulikakos and Bejan [11] established a theoretical framework for the minimization of entropy generation in extended surfaces. They derived the entropy generation formula for a general fin geometry and presented specific results for optimum dimensions of fins with various cross-sections. The study of Cheng et al. [12] was concerned with local entropy generation of the laminar mixed-convection flow in a vertical channel with a series of transverse fins. Results for various physical and geometric parameters were presented and the geometric configuration with higher second-law efficiency was proposed. Second law analysis on a pin-fin array under crossflow was conducted by Lin and Lee [13], from which the entropy generation rate was evaluated. In their study optimum design/operational conditions were searched for on the basis of entropy generation minimization. Comparison
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between the staggered and the in-line pin-fin alignments were also reported. Thermoeconomic optimization of constant cross-sectional area fins was presented by Shuja and Zubair [14]. They provided an analytical approach for the optimized design while considering the material cost. In a recent work, Culham and Muzychka [15] presented a procedure that allowed simultaneous optimization of heat sink design parameters based on minimization of the entropy generation associated with heat transfer and fluid friction. They considered all relevant design parameters for plate fin heat sinks, including geometric parameters, heat dissipation, material properties and flow conditions, to characterize a heat sink that minimized entropy generation and in turn resulted in a minimum operating temperature. In the present study optimum geometry of a pin fin array is obtained based on exergoeconomic analysis resulting in a minimum cost of operation, while considering both in-line and staggered arrangements. Results are given in analytical form through equations as well as in graphical form.
CD = Cm Rem D,
2. Thermoeconomic analysis and optimization
Nu = Cn RenD Pr1/3
(4)
Eq. (2) can be written as
The arrangement of the in-line and staggered fin array considered, is schematically shown in Fig. 1. In the following analysis, an objective function which relates the cost of operation (Γ˙ ) to the system irreversibility is developed. The objective function can be written as [14]: Γ˙ = λH S˙T + λP S˙P
equation (1), we get the cost of operation in terms of thermal, physical and unit cost parameters, which is given by: kf √ kf √ ReL π Nu tanh 2 Nu NV Γ˙ = λH F ReD 2 ks ks ReD −1 kf + Nu G1 ks N + λP F B ReL ReD CD G2 (2) 2 where N and V are the number of rows and number of columns, respectively in the fin array. Other dimensionless parameters F , B, G1 and G2 are defined in Appendix A according to Eqs. (A.8) and (A.9). Introducing the dimensionless parameter M as: ks /kf M= (3) Pr1/6 and the following functional forms for CD and Nu with the constants given in Table 1,
(1)
where λH and λP are the costs associated with heat transfer and pressure drop irreversibilities, respectively. The unit costs represent the income lost for the overall system. The various methods of calculating λH and λP are reviewed by Ranasinghe et al. [16]. Derivation of the entropy generation due to heat transfer (S˙T ) and due to pressure drop (S˙P ) is given in Appendix A through Eqs. (A.10) and (A.11), respectively. On substituting these relations for the entropy generation in the cost
Γ˙ = λH F M 2 √ Cn n/2−1 π n/2+1 NV M Cn ReD ReD × tanh 2 ReL 2 M −1 + Cn Ren+1 G 1 D + λP F
N BCm Rem+1 D ReL G2 2
(5)
Here Γ˙ , apart from other variables is a function of length (L) and diameter (D) which are expressed in terms of ReL and ReD . 2.1. Minimum cost based on optimum ReL The first step in calculating optimum Γ˙ , is to differentiate Eq. (5) with respect to ReL , and set it to zero. The
Fig. 1. Schematic view of the two arrangements of pin-fin array.
S.Z. Shuja / Exergy, an International Journal 2 (2002) 248–258
251
Table 1 Coefficient for Eqs. (4) [17,18] ReD = Umax D/ν
Staggered arrays
10–100 100–1000 1000–2 × 105 2 × 105 –106
In-line arrays
Cn
n
Cn
n
0.8 0.71 0.35(Sp /Sn )−0.2 for (Sp /Sn ) < 2 0.4 for (Sp /Sn ) > 2 0.031(Sp /Sn )−0.2
0.4 0.5 0.6
0.9 0.52 0.27
0.4 0.5 0.63
0.88
0.03
0.8
Cm
m
0.47 1.0 + (Sn /D − 1)1.08
−0.155
differentiation becomes convenient if we rewrite Eq. (5) in the following form: 1 ˙ Γ =W + Z ReL (6) X tanh(Y ReL ) + 1 W , X, Y and Z are functions of ReD and are given as −n/2
, W = CRe−n−1 D n/2−1
Y = C2 ReD
X = C1 ReD
Z = C3 Rem+n+2 D
,
(7)
where λH F M 2 , Cn G1 √ 2 Cn C2 = , M C=
C1 = C3 =
π NV M √ 2 G1 Cn
B N λP Cm Cn G1 G2 2 M 2 λH
(8)
Then Eq. (6) can also be written as: Γ˙ ∗ = =
Γ˙ (ReL , ReD ) C
0.176 +
Cm 0.32(Sp /Sn )(Sn /D) (Sn /D − 1)0.43+1.13 (D/Sn )(Sn /Sp )
m −0.155
√ √ √ Noting that cosh−1 ( x ) = tanh−1 ( x − 1/ x ), Eq. (11) can also be rewritten as 1 1 −1 (13) 1− ReLopt = tanh Y x On substituting the value of ReLopt from the above equation, we can now express Eq. (9) as Γ˙ (ReLopt , ReD ) Re−n−1 D = −n/2 √ C C1 ReD 1 − 1/x + 1 √ C3 m−n/2+2 Re cosh−1 x (14) + C2 D where C, C1 , C2 and C3 are constants as defined in Eq. (8). Eqs. (11) and (14) are useful when the fin dimension D is fixed due to some constraint and it is only required to find the optimum with respect to the dimension L; then Eq. (11) gives the optimum length and Eq. (14) gives the corresponding optimum cost. It should be noted that the dimension (D) is reflected through the variables (x, W , X, Y and Z) which are functions of ReD as shown in Eqs. (12) and (7).
1 n/2+1 n/2−1 C1 ReD tanh(C2 ReD ReL ) + Ren+1 D + C3 Rem+1 Re L D
2.2. Minimum cost based on optimum ReL and ReD (9)
Differentiating Eq. (6) with respect to ReL and equating to zero, results in: 1 ∂ Γ˙ −XY = + Z = 0 (10) W ∂ReL [cosh(Y ReL ) + X sinh(Y ReL )]2 by applying some algebraic rules this equation can be solved for ReL , which gives, ReLopt =
√
1 cosh−1 x Y
(11)
where x is given by x = f (ReD ) 2
X = Y X + 1 + XZ X2 − 1 2 2 Z(X − 1)
− 2 XY XY + Z X2 − 1
(12)
As a second step towards optimization we now find the derivative of Eq. (14) with respect to ReD . dΓ˙ dReD
1 + n + (n/2 + 4)C4 1 − 1/x Re−n−2 D −n/2+1 C1 Re dx × √ D 2x x(x − 1) dReD
2 −1 CC3 m−n/2+2 −n/2 × 1 + C1 ReD 1 − 1/x + ReD C2 √ dx (m − n/2 + 2) cosh−1 x 1 × − √ ReD 2 x(x − 1) dReD XReD dx 1− √ 2x x(x−1) dReD √ =W (1 + X 1 − 1/x )2 √ Z cosh−1 x 1 dx + − √ (15) Y ReD 2 x(x − 1) dReD
= −CRe−n−2 D
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Differentiating Eq. (12) with respect to ReD gives C1 Re−m−4 dx D = dReD C3 (RenD − C12 )3 × C14 C2 (m + 3) − nC12 (3C2 + K)RenD C1 C2 1−n/−1 ReD + (nK − pC2 )Re2n D + J × C13 C2 (3n − 2p) + C1 C2 (n + 2p)RenD Ren + K C1 − D C1 2
× C1 (2n − p) + (n + p)RenD where K = C1 C2 Rem+3 and D C1 (K + C2 ) J = XY −Z and ReD p=m+n+3 substituting the above equation in Eq. (15), we get 1 dΓ˙ = f (ReD , C1 , C2 , C3 ) C dReD
(16)
The minimum operational cost is found when Eq. (16) is equated to zero, this gives f (ReD , C1 , C2 , C3 ) = 0
(17)
It should be noted that m and n are known values for a given fin array and depend on ReD . Eq. (17) is a transcedental equation which can be solved for the optimum ReD , so that as a solution of this equation we can write ReDopt , = g(C1 , C2 , C3 )
(18)
This optimum value of (ReDopt ) if substituted in Eq. (11), will give optimum (ReLopt ) and if used in Eq. (14) will give the minimum cost of operation (Γ˙opt).
3. Results and discussion In the solution presented above C1 , C2 and C3 are recognized as the three dimensionless parameters influencing the cost of operation (Γ˙ ) for a pin-fin array (according to Eq. (9)) and the √ optimum fin geometry (see Eqs. (11) and (18)). C2 = 2 Cn /M is related to the fin material and the cooling fluid. For a typical application, using air as the cooling fluid (kf = 0.0265 W·m−1 ·K−1 ; Pr = 0.716) and Copper as the fin material (ks = 400 W·m−1 ·K−1 ), we get M = 130 (Eq. (3)), and C2 = 0.011 and 0.01455 for in-line and staggered arrangements, respectively. On the other hand if aluminium (ks = 237 W·m−1 ·K−1 ) is used as the fin material then M = 100, and C2 = 0.01437 and 0.0189 for inline and staggered arrangements, respectively. Therefore to
cover both Aluminium and Copper fin arrays, a range for C2 = 0.01–0.02 is chosen for the presented results. √ A parametric study for C1 = (π/2)(NV M/(G1 Cn )) is also performed to obtain a practical range. Considering variation of N and V from (3–30), M from (100–130), Sp /D and Sn /D from (1.25–4) and Cn from (0.27–0.9) yields a range of C1 = 11.5–682.6. Thus, a range for C1 = 10–1000 is chosen for the presented results. Parametric study for C3 = (B/2)(N/M 2 )(λP /λH )Cm × Cn G1 G2 indicated a wide range because of the large possible variations of B and λP /λH , the range for C3 = 10−12 –10−8 was selected for presenting results in graphical form. Fig. 2 shows the plot of dimensionless cost of operation (Γ˙ ∗ ) against ReD and ReL (Eq. (9)) for C1 = 1000, C2 = 0.015 and C3 = 10−11 . It indicates that there exist a point for minimum cost in the range of ReD and ReL shown. The minimum dimensionless cost was found as 4.32947 × 10−6 for ReD = 200.44 and ReL = 2159.83. Fig. 3 shows the plot of Eq. (14) which is the dimensionless cost obtained after the first step of optimization so that ReLopt (from Eq. (11)) has been substituted for ReL . The plot is shown for C1 = 1000, C2 = 0.015, and C3 = 10−11 and staggered arrangement. Two points of minimum A and B can be seen in the plot. Point A corresponds to ReD,A = 200.44 and point B corresponds to ReD,B = 3784.53. Using Eq. (11), ReLopt ,A = 2159.83 and ReLopt ,B = 40.056 and (L/D)A = 10.78 and (L/D)B = 0.0106. It is clear that L/D = 0.0106 is not a practical result although it corresponds to an overall minimum. Based on this, points similar to A which are local minimum in the range of L/D 1 were selected for determining ReDopt . Thus all the results presented below are subjected to the constraint that L/D is either of the order of 1 or greater than 1. Fig. 4 shows the plot of optimum Reynolds number (ReDopt ) against C3 for various values of C1 and C2 using Eq. (18). It can be seen that as C3 increases ReDopt decreases to low values which is reasonable, since low C3 implies small values of the fluid friction irreversibility coefficient as can be seen from the dimensionless variable B. In addition if (λP /λH ) is large, i.e., unit cost of pressure drop is more than unit cost of heat transfer, then also it is more desirable to make fins with smaller diameter (i.e., thin fins). The effect of C1 on ReDopt can also be seen from Fig. 4. As C1 increases (e.g., an array with more number of fins) the diameter of each fin should reduce for optimum cost of operation. The √ effect of C2 = 2 Cn /M is not as significant as C1 and C3 on ReDopt in the presented practical range of the values of C2 from 0.01–0.02.The small variation shows that as C2 increases (i.e., M = ks /kf /Pr1/6 becomes smaller) ReDopt also increases. The sudden variation in the results for ReDopt around ReD = 100 and ReD = 1000 for all results shown in Fig. 4 is due to different values of n in the different ranges of ReD , as shown in Table 1. Fig. 5 shows the variation of ReLopt with C1 , C2 and C3 . The effect of C3 on ReLopt is similar to the effect of C3
S.Z. Shuja / Exergy, an International Journal 2 (2002) 248–258
253
Fig. 2. Plot of the dimensionless cost of operation (Γ˙ ∗ ) against ReD and ReL for a pin-fin array showing the point of minimum cost of operation.
Fig. 3. Plot of dimensionless cost of operation (Γ˙ ∗ ) and the corresponding L/D ratio against ReD showing two minimums A and B, and the range for which L/D is > 1.
on ReDopt . As C3 increases ReLopt decreases for all ranges of C1 and C2 considered in this study. However as C1 increases from 10–100, ReLopt increases and then decreases for C1 = 1000. Fig. 6 shows the plot of L/D ratio against C3 for different values of C1 and C2 . For C1 = 10, L/D is of the order of 1, since ReDopt and ReLopt are almost same. In
this case, for lower values of C2 , L/D even becomes lower than 1. For C1 = 100 and 1000, L/D is higher (varies in the range 5–40) with the variation of C3 from 10−12 –10−8 , since for these values ReDopt decreases but ReLopt increases. Fig. 7 shows the dimensionless optimum cost of opera∗ for different values of C , C and C . As C intion Γ˙opt 1 2 3 3
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S.Z. Shuja / Exergy, an International Journal 2 (2002) 248–258
Fig. 4. Plot of optimum Reynolds number (ReDopt ) against C3 for various values of C1 and C2 .
Fig. 5. Plot of optimum Reynolds number (ReLopt ) against C3 for various values of C1 and C2 .
creases the cost of operation also increases for all values of C1 and C2 considered, which is what is anticipated, because increase of C3 implies large fluid friction irreversibility coefficient B (refer to Eq. (A.8)) and small M (fins with smaller ks /kf ratio). The figure also shows that as C1 decreases (less no. of fins), the optimum cost of operation also decreases, reaching a minimum for large C1 . The variable C2 has less effect on the optimum cost for the range of values of C2 considered in the present study. The sudden variation in cost appearing in Fig. 7 is due to the sudden change in parameter n for the various ranges of ReD as shown in Table 1. It should be noted that for various ranges of the ReD appearing in the results, the parameter n is same for staggered and in-line fin arrangement (see Table 1). Thus
all the figures presented above are valid for both staggered and in-line arrangements. It should, however, be noted that ∗ would be different for the results for ReDopt , ReLopt and Γopt the two arrangements because C1 , C2 and C3 are different for the two cases as shown in the example, discussed below. Illustrative example: To explain the methodology presented in the present study, we consider the case of an aluminium 10 × 10, pin-fin array (ks = 237 W·m−2 ·K−1 ) which is to be optimized for usage in electronics cooling. The cooling fluid is air at an average temperature of 300 K (ρ = 1.1644 kg·m−3 ; ν = 1.6 × 10−5 m2 ·s−1 ; kf = 0.0265 W·m−2 ·K−1 ; Pr = 0.712) with the maximum flow velocity, Umax = 1 m·s−1 and a heat transfer rate of 0.5 W
S.Z. Shuja / Exergy, an International Journal 2 (2002) 248–258
Fig. 6. Plot of L/D against C3 for various values of C1 and C2 .
∗ ) against C for various values of C and C . Fig. 7. Plot of optimum dimensionless cost (Γ˙opt 3 1 2
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Table 2 Calculated values of hydraulic, thermal and geometric coefficients Cn (Table 1)
Cm (Table 1)
G1 (Eq. (A.9))
G2 (Eq. (A.9))
0.52 0.71
1.02951 1.74843
172.683 185.759
5.850 6.175
In-line Staggered
Table 3 Calculated values of thermoeconomic coefficients λ F M2
C= H Cn G 1
C2 = 2 MCn
λ C3 = B2 N2 λ P Cm Cn G1 G2 M H
126.145 100.355
0.014422 0.016852
3.6677 × 10−11 9.6572 × 10−11
G 1 Cn
8.158 × 10−4 5.554 × 10−4
In-line Staggered
√
C1 = π2 NV√M
Table 4 Final solution of optimum geometric parameters W
X
1.6669 × 10−7 1.4460 × 10−7
30.6145 25.3591
ReDopt Eq. (18) or Fig. 4 In-line Staggered
Y
Z
2.062 × 10−4 2.719 × 10−4
2.15 × 10−5 3.88 × 10−5
Eq. (7)
288 245
Table 5 Final solution of optimum geometric and cost values
In-line Staggered
x
ReLopt
Γ˙opt (ReLopt , ReDopt ) $·yr−1
Eq. (12)
Eq. (13) or Fig. 5
Eq. (14) or Fig. 7
1.27324 1.23243
2432.3 1710.6
0.6212 0.6820
for the fin array. The unit cost parameter associated with the heat transfer irreversibility, λH = 0.001$·K·J−1 ), and the cost ratio λP /λH = 0.1. The fin spacing ratio Sn /D = 1.65 and Sp /Sn = 1. Solution: The entropy parameter (F ) and the dimensionless parameter that accounts for the importance of fluid friction irreversibility relative to heat irreversibility (B) are q 2 Umax F = B 2 = 0.007325 W·K−1 νks T∞ B=
ρν 3 k
s T∞ 2 qB
= 1.3564 × 10−9
while the material parameter M is ks /kf M= = 100 Pr1/6 The other dimensionless parameters are calculated in Table 2. Now using Eq. (8) the solution for dimensionless parameters are in Table 3. Furthermore, the final solution can be tabulated in Tables 4 and 5. It can be seen from the above results that for similar thermoeconomic operating conditions, Dopt and Lopt for
Dopt (mm)
Lopt (mm)
4.608 3.920
38.92 27.37
the in-line arrangement of the fins is about 17% and 42%, respectively, more than the staggered configuration. In addition, although the cost of operation for in-line arrangement is less than that corresponding to the staggered arrangement, but the size and, therefore, corresponding material cost are more for the in-line arrangement.
4. Conclusions Optimum geometry of a pin fin array (with diameter and length as variables) is obtained by exergoeconomic analysis. Both in-line and staggered arrangements are studied. Practical ranges of the three solution parameters C1 , C2 and C3 defined through Eq. (8) are considered. Results are presented in analytical form for ReDopt , ReLopt and Γ˙opt through Eqs. (18), (13) and (14), respectively. Graphical solutions are also given for approximate handy calculations and fast interpretation. The parameter C3 representing fluid friction irreversibility coefficient B and unit cost ratio (λP /λH ) as well as the parameter C1 representing the number of fins and fin geometry, both affect the results such that the minimum cost increases as C3 increases and decreases as C1 increases. In addition, the optimum Reynolds number ReDopt decreases
S.Z. Shuja / Exergy, an International Journal 2 (2002) 248–258
with increase in C3 and C1 . C2 which has a practical range of 0.01–0.02 does not significantly affect the minimum cost of operation as well as ReDopt , although it is shown to have more affect on ReLopt . It is important to note that the presented results are more relevant where the operation is to be in the low ranges of Reynolds number, which is practical for cooling of electronic components, particularly in the mixed convection or purely natural convection regimes.
Acknowledgements The author acknowledge the support provided by King Fahd University of Petroleum and Minerals for this research project.
257
D2 2 π NV kf mf L tanh(mf L) S˙gen = qB2 T∞ 4 L −1 π 2 + hw A − NV D 4 3 L(V − a)(S − D) CD NρUmax n (A.6) 2T∞ After non-dimensionalizing the diameter D as ReD and the length L as ReL , and assuming that hw = hf = h, the above equation can be rewritten as kf √ kf √ ReL π ˙ NV Nu tanh 2 Nu Sgen = F ReD 2 ks ks ReD −1 kf N + Nu G1 + F BReL ReD CD G2 ks 2 (A.7)
+
where Appendix A. Second-law analysis for a pin-fin array The irreversible losses of a fin array have been shown to consist mainly of those due to heat transfer and pressure drop. Thus the traditional irreversibility rate based objective function for a fin can be expressed as [13], qB θB mP ˙ S˙gen = 2 + ρT∞ T∞
(A.1)
In the above equation the term on the right-hand side indicate, the entropy generated due to heat transfer, and fluid friction. In this solution it has been assumed that the temperature difference, θB = TB − T∞ , is small compared with the absolute temperature, i.e., θB T∞ . According to the unidirectional heat conduction model for a fin array with constant cross-section area fins, the base stream temperature difference is given by [19], D2 π NV kf (mf L) tanh(mf L) θB = qB 4 L π + hw A − NV D 2 (A.2) 4 where mf =
F=
4hf ks D
B=
ρν 3 ks T∞ qB2
and the dimensionless geometric parameters are Sp Sn +1 G1 = (N − 1) Sn D π Sn × (V − a) + 1 − NV D 4 Sn G2 = (V − a) −1 D
(A.8)
(A.9)
In Eq. (A.7), the entropy generation is a summation of two terms which can be separated as S˙T and S˙P ; i.e., entropy generation due heat transfer and pressure drop, respectively, as kf √ π NV Nu S˙T = F ReD 2 ks −1 kf kf √ ReL Nu × tanh 2 + Nu G1 ks ReD ks (A.10) S˙P
qB2 Umax , 2 νks T∞
N = F B ReL ReD CD G2 2
(A.11)
and
A = (N − 1)Sp + D (V − a)Sn + D
(A.3)
Also, the mass flow rate is defined as m ˙ = ρUmax L(V − a)(Sn − D)
(A.4)
where Umax is the maximum average fluid velocity occurring at the minimum free flow area of the fin array. The pressure drop is defined using the drag coefficient, as [19] 1 2 ρU P = CD N (A.5) 2 max Using Eqs. (A.2), (A.4) and (A.5) in Eq. (A.1), we get
References [1] G. Tsatsaronis, Exergoeconomics—What is it?, in: Energy Conversion and Protection of the Environment, Institute for Energy Engineering, http://www.iet.tu-berlin.de/efeu/exergo.htm. [2] A. Bejan, G. Tsatsaronis, M. Moran, Thermal Design and Optimization, Wiley, New York, 1996. [3] K.S. Lau, R.L. Mahajan, Effects of tip clearance and fin density on the performance of heat sinks for VLSI packages, IEEE Trans. Components, Hybrids, Manufacturing Technol. 12 (4) (1989) 757– 765. [4] S. Lee, Optimum design and selection of heat sinks, IEEE Trans. Components, Packaging, Manufacturing Technol. Part A 18 (4) (1995) 812–817.
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[5] A. Guvenc, H. Yuncu, An experimental investigation on performance of fins on a horizontal base in free convection heat transfer, Heat Mass Transfer 37 (4–5) (2001) 409–416. [6] B. Pierluigi, S. Enrico, Numerical calculation of the local rate of entropy generation in the flow around a heated finned-tube, Thermodyn. Design, Anal. Improvement Energy Syst. 30 (1993) 81–91. [7] K.A. Moores, Y.K. Joshi, G.H. Schiroky, Thermal characterization of a liquid cooled AlSiC base plate with integral pin fins, IEEE Trans. Components, Packaging Technol. 24 (2) (2001) 213–219. [8] R.W. Knight, J.S. Goodling, B.E. Gross, Optimal thermal design of air cooled forced convection finned heat sinks—Experimental verification, IEEE Trans. Components, Hybrids, Manufacturing Technol. 15 (5) (1992) 754–760. [9] M. Iyengar, A. Bar-Cohen, Optimization of vertical pin-fin heat sinks in natural convective heat transfer, in: Proceedings of the 11th International Heat Transfer Conference (IHTC), Vol. 3, Kyongju, Korea, 1998, pp. 501–506. [10] A.T. Morrison, Optimization of heat sink fin geometries for heat sinks in natural convection, in: InterSociety Conference on Thermal Phenomena, IEEE, 1992, pp. 145–148. [11] D. Poulikakos, A. Bejan, Fin geometry for minimum entropy generation in forced convection, ASME J. Heat Transfer 104 (1982) 616–623.
[12] H. Cheng, W. Ma, Numerical predictions of entropy generation for mixed convective flows in a vertical channel with transverse fin array, Internat. Commun. Heat Mass Transfer 21 (4) (1994) 519–530. [13] W.W. Lin, D.J. Lee, Second-law analysis on a pin-fin array under crossflow, Internat. J. Heat Mass Transfer 40 (8) (1997) 1937–1945. [14] S.Z. Shuja, S.M. Zubair, Thermoeconomic optimization of constant cross-sectional area fins, ASME J. Heat Transfer 119 (4) (1997) 860–863. [15] J.R. Culham, Y.S. Muzychka, Optimization of plate fin heat sinks using entropy generation minimization, IEEE Trans. Components, Packaging Technol. 24 (2) (2001) 159–165. [16] J. Ranasinghe, S. Aceves-Saborio, G.M. Reistad, Optimization of heat exchangers in energy conversion systems, in: M.J. Moran, E. Sciubba (Eds.), Second Law Analysis of Thermal Systems, ASME, New York, 1987, pp. 29–38. [17] A. Zukauskas, R. Ulinskas, Efficiency parameters for heat transfer in tube banks, Heat Trasfer Engrg. 6 (1985) 19–25. [18] A.J. Chapman, Fundamentals of Heat Transfer, Macmillan, New York, 1987. [19] F.P. Incropera, D.P. Dewitt, Introduction to Heat Transfer, Wiley, New York, 1985.
Optimal fin geometry based on exergoeconomic analysis for a pin-fin array with application to electronics cooling S.Z. Shuja Department of Mechanical Engineering, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia Received 6 May 2002; accepted 1 June 2002
Abstract Exergoeconomic analysis for a pin-fin array involves the achievement of a balance between the entropy generation due to heat transfer and pressure drop, while considering the unit cost of entropy generation. This process yields the optimum fin operation parameters based on minimum cost. In this study, analytical equations are presented considering the cost of operation for a pin-fin array. The solution of these equations would give the optimum fin diameter and length that result in a fin array with minimum operational cost. In addition, the influence of important fin thermal, physical, geometrical and cost parameters on the optimum diameter and length is presented in graphical form for quick calculations and easy interpretation. The presented results are subjected to the constraint that L/D is of the order of 1 or greater than 1. A case is also presented to demonstrate the use of the model for conditions typically found in cooling of electronic components. 2002 Éditions scientifiques et médicales Elsevier SAS. All rights reserved. Keywords: Thermoeconomic; Fin array; Optimization; Electronics cooling
1. Introduction Heat transfer enhancement devices are often employed to increase the rate of heat transfer. For applications in electronics cooling, the objective is to maintain the operating temperatures at a safe level for long term, reliable operation. In most cases, however, the selection of the most appropriate geometry for a particular application is difficult, since by adding a heat transfer enhancement device, not only the rate of heat transfer increases (reducing the heat transfer irreversibilities), but also the fluid friction increases (increasing the hydrodynamic irreversibilities). This raises the question as to what is the real advantage of the employed enhancement technique. To answer this question the well-known exergoeconomic analysis can be applied to the heat transfer enhancement devices. As described by Tsatsaronis [1], the exergoeconomic (thermoeconomic) analysis, evaluation, and optimization techniques are based on a combination of: (a) an exergy analysis, (b) a conventional economic analysis to calculate levelized costs, and (c) application of exergy costing, which recognizes that exergy provides the only rational basis for assigning costs to energy systems and energy carriers. The E-mail address: [email protected] (S.Z. Shuja).
exergoeconomic analysis identifies the cost associated with the real thermodynamic waste (exergy destruction and exergy loss) at the component level and reveals the real cost sources for each component. A comparison of the cost of exergy destruction within the component of a system with the component investment, operation, and maintenance costs for each component provides useful information for improving the cost effectiveness of the component as well as the overall system. An exergoeconomic analysis is very useful, particularly for complex thermal systems, in optimizing the overall system parameters or specific variables in a single component. A detailed description of the exergoeconomic analysis, evaluation, and optimization technique can be found in [2]. The continuous miniaturization of microelectronics, is posing new challenges in electronic cooling [3]. This initiated a wide research in the optimization of the heat-sink design. An overview of different types of heat sinks and associated design parameters are provided by Lee [4]. He also developed an analytical model for predicting and optimizing the thermal performance of bi-directional fin heat sinks in a partially confined configuration. Lau and Mahajan [3] performed experiments to examine the heat transfer and pressure drop characteristics for several heat sink geometries at various mass flow rates. They indicated that although the heat transfer increased as the fin density was increased, ac-
1164-0235/02/$ – see front matter 2002 Éditions scientifiques et médicales Elsevier SAS. All rights reserved. PII: S 1 1 6 4 - 0 2 3 5 ( 0 2 ) 0 0 0 8 1 - X
S.Z. Shuja / Exergy, an International Journal 2 (2002) 248–258
249
Nomenclature a
fin array geometric parameter (1 for inline; 1/2 for staggered) A flow area defined in Eq. (A.3) . . . . . . . . . . . . . m2 B fluid friction irreversibility coefficient, Eq. (A.8) C, C1 , C2 , C3 parameters defined by Eq. (8) CD drag coefficient, Eq. (4) constant in Eq. (4) Cm Cn constant in Eq. (4) D diameter of the pin fin . . . . . . . . . . . . . . . . . . . . m F parameter defined by Eq. (A.8) G1 , G2 geometric parameters defined by Eq. (A.9) hf , hw , h heat transfer coefficient . . . . . . . . W·m−2 ·K−1 kf thermal conductivity of fluid . . . . . W·m−1 ·K−1 ks thermal conductivity of fin material . . . . . . . . . . . . . . . . . . . . . . . W·m−1 ·K−1 L length of the pin fin . . . . . . . . . . . . . . . . . . . . . . . m m constant in Eq. (4) m ˙ mass flow rate defined by Eq. (A.4) . . . . kg·s−1 mf fin constant defined by Eq. (A.3) M constant defined by Eq. (3) n constant in Eq. (4) N number of rows in a fin array Nu Nusselt number Pr Prandtl number P pressure drop . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pa qB base heat transfer . . . . . . . . . . . . . . . . . . . . . . . . W Reynolds number based on diameter ReD
companying this increased heat transfer was an increase in the pressure drop. Natural convection heat transfer in rectangular fin-arrays mounted on a vertical base was investigated experimentally by Guvenc and Yuncu [5]. Their results showed that there existed an optimum fin spacing for which the heat transfer rate from the fin array was maximized. Pierluigi and Enrico [6] performed numerical calculations for the local entropy generation in the flow around a heated finned tube. They determined optimal spacing of fins using entropy generation rate as well as total heat transfer as objective functions. Moores et al. [7] employed computational fluid dynamics and heat transfer techniques to model the thermal and hydrodynamic resistance characteristics through the pinfin structure of a prototype base plate design. They also verified the performance experimentally using water as the cooling fluid. Knight et al. [8] used an optimization scheme to design several air cooled aluminum finned arrays. They built and tested these designs experimentally and verified that the design obtained actually yield lowest thermal resistance, and hence, desirable operating temperatures, when used with the specified constraints. The optimal spacing and diameter for an isothermal fin array of fixed volume, appropriate for use with electronic equipment, was determined by Iyengar and
Reynolds number based on length entropy generation rate . . . . . . . . . . . . . . . W·K−1 entropy generation rate due to pressure drop, Eq. (A.10) . . . . . . . . . . . . . . . . . . . . . . . . . . W·K−1 ˙ ST entropy generation rate due to heat transfer, Eq. (A.11) . . . . . . . . . . . . . . . . . . . . . . . . . . W·K−1 Sn pin spacing in spanwise direction . . . . . . . . . . m Sp pin spacing in streamwise direction . . . . . . . . m T∞ absolute temperature of free stream . . . . . . . . . K maximum average fluid velocity occuring at the Umax minimum free flow area . . . . . . . . . . . . . . . m·s−1 V number of columns in the fin array W , X, Y , Z variables defined by Eq. (7) x variable defined by Eq. (12)
ReL S˙gen S˙P
Greek symbols θB ρ ν λH λP Γ˙ Γ˙ ∗
base stream temperature difference, = TB − T∞ density of fluid . . . . . . . . . . . . . . . . . . . . . . kg·m−3 kinematic viscosity of fluid . . . . . . . . . . . m2 ·s−1 unit cost of lost work due to heat transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . $·K·J−1 unit cost of lost work due to pressure drop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . $·K·J−1 cost of operation . . . . . . . . . . . . . . . . . . . . . . $·s−1 dimensionless cost of operation, = Γ˙ /C
Subscript opt
optimum
Bar-Cohen [9]. They discussed the influence of the pin-fin length on the optimal horizontal spacing and diameter, and the optimal heat dissipation with reference to a specified heat sink configuration. Morrison [10], presented the optimization of fin geometry for heat sinks in natural convection with rectangular cross-section fins at a constant fin spacing, and applied the method for steady state and intermittent duty cycle operation. Poulikakos and Bejan [11] established a theoretical framework for the minimization of entropy generation in extended surfaces. They derived the entropy generation formula for a general fin geometry and presented specific results for optimum dimensions of fins with various cross-sections. The study of Cheng et al. [12] was concerned with local entropy generation of the laminar mixed-convection flow in a vertical channel with a series of transverse fins. Results for various physical and geometric parameters were presented and the geometric configuration with higher second-law efficiency was proposed. Second law analysis on a pin-fin array under crossflow was conducted by Lin and Lee [13], from which the entropy generation rate was evaluated. In their study optimum design/operational conditions were searched for on the basis of entropy generation minimization. Comparison
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between the staggered and the in-line pin-fin alignments were also reported. Thermoeconomic optimization of constant cross-sectional area fins was presented by Shuja and Zubair [14]. They provided an analytical approach for the optimized design while considering the material cost. In a recent work, Culham and Muzychka [15] presented a procedure that allowed simultaneous optimization of heat sink design parameters based on minimization of the entropy generation associated with heat transfer and fluid friction. They considered all relevant design parameters for plate fin heat sinks, including geometric parameters, heat dissipation, material properties and flow conditions, to characterize a heat sink that minimized entropy generation and in turn resulted in a minimum operating temperature. In the present study optimum geometry of a pin fin array is obtained based on exergoeconomic analysis resulting in a minimum cost of operation, while considering both in-line and staggered arrangements. Results are given in analytical form through equations as well as in graphical form.
CD = Cm Rem D,
2. Thermoeconomic analysis and optimization
Nu = Cn RenD Pr1/3
(4)
Eq. (2) can be written as
The arrangement of the in-line and staggered fin array considered, is schematically shown in Fig. 1. In the following analysis, an objective function which relates the cost of operation (Γ˙ ) to the system irreversibility is developed. The objective function can be written as [14]: Γ˙ = λH S˙T + λP S˙P
equation (1), we get the cost of operation in terms of thermal, physical and unit cost parameters, which is given by: kf √ kf √ ReL π Nu tanh 2 Nu NV Γ˙ = λH F ReD 2 ks ks ReD −1 kf + Nu G1 ks N + λP F B ReL ReD CD G2 (2) 2 where N and V are the number of rows and number of columns, respectively in the fin array. Other dimensionless parameters F , B, G1 and G2 are defined in Appendix A according to Eqs. (A.8) and (A.9). Introducing the dimensionless parameter M as: ks /kf M= (3) Pr1/6 and the following functional forms for CD and Nu with the constants given in Table 1,
(1)
where λH and λP are the costs associated with heat transfer and pressure drop irreversibilities, respectively. The unit costs represent the income lost for the overall system. The various methods of calculating λH and λP are reviewed by Ranasinghe et al. [16]. Derivation of the entropy generation due to heat transfer (S˙T ) and due to pressure drop (S˙P ) is given in Appendix A through Eqs. (A.10) and (A.11), respectively. On substituting these relations for the entropy generation in the cost
Γ˙ = λH F M 2 √ Cn n/2−1 π n/2+1 NV M Cn ReD ReD × tanh 2 ReL 2 M −1 + Cn Ren+1 G 1 D + λP F
N BCm Rem+1 D ReL G2 2
(5)
Here Γ˙ , apart from other variables is a function of length (L) and diameter (D) which are expressed in terms of ReL and ReD . 2.1. Minimum cost based on optimum ReL The first step in calculating optimum Γ˙ , is to differentiate Eq. (5) with respect to ReL , and set it to zero. The
Fig. 1. Schematic view of the two arrangements of pin-fin array.
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251
Table 1 Coefficient for Eqs. (4) [17,18] ReD = Umax D/ν
Staggered arrays
10–100 100–1000 1000–2 × 105 2 × 105 –106
In-line arrays
Cn
n
Cn
n
0.8 0.71 0.35(Sp /Sn )−0.2 for (Sp /Sn ) < 2 0.4 for (Sp /Sn ) > 2 0.031(Sp /Sn )−0.2
0.4 0.5 0.6
0.9 0.52 0.27
0.4 0.5 0.63
0.88
0.03
0.8
Cm
m
0.47 1.0 + (Sn /D − 1)1.08
−0.155
differentiation becomes convenient if we rewrite Eq. (5) in the following form: 1 ˙ Γ =W + Z ReL (6) X tanh(Y ReL ) + 1 W , X, Y and Z are functions of ReD and are given as −n/2
, W = CRe−n−1 D n/2−1
Y = C2 ReD
X = C1 ReD
Z = C3 Rem+n+2 D
,
(7)
where λH F M 2 , Cn G1 √ 2 Cn C2 = , M C=
C1 = C3 =
π NV M √ 2 G1 Cn
B N λP Cm Cn G1 G2 2 M 2 λH
(8)
Then Eq. (6) can also be written as: Γ˙ ∗ = =
Γ˙ (ReL , ReD ) C
0.176 +
Cm 0.32(Sp /Sn )(Sn /D) (Sn /D − 1)0.43+1.13 (D/Sn )(Sn /Sp )
m −0.155
√ √ √ Noting that cosh−1 ( x ) = tanh−1 ( x − 1/ x ), Eq. (11) can also be rewritten as 1 1 −1 (13) 1− ReLopt = tanh Y x On substituting the value of ReLopt from the above equation, we can now express Eq. (9) as Γ˙ (ReLopt , ReD ) Re−n−1 D = −n/2 √ C C1 ReD 1 − 1/x + 1 √ C3 m−n/2+2 Re cosh−1 x (14) + C2 D where C, C1 , C2 and C3 are constants as defined in Eq. (8). Eqs. (11) and (14) are useful when the fin dimension D is fixed due to some constraint and it is only required to find the optimum with respect to the dimension L; then Eq. (11) gives the optimum length and Eq. (14) gives the corresponding optimum cost. It should be noted that the dimension (D) is reflected through the variables (x, W , X, Y and Z) which are functions of ReD as shown in Eqs. (12) and (7).
1 n/2+1 n/2−1 C1 ReD tanh(C2 ReD ReL ) + Ren+1 D + C3 Rem+1 Re L D
2.2. Minimum cost based on optimum ReL and ReD (9)
Differentiating Eq. (6) with respect to ReL and equating to zero, results in: 1 ∂ Γ˙ −XY = + Z = 0 (10) W ∂ReL [cosh(Y ReL ) + X sinh(Y ReL )]2 by applying some algebraic rules this equation can be solved for ReL , which gives, ReLopt =
√
1 cosh−1 x Y
(11)
where x is given by x = f (ReD ) 2
X = Y X + 1 + XZ X2 − 1 2 2 Z(X − 1)
− 2 XY XY + Z X2 − 1
(12)
As a second step towards optimization we now find the derivative of Eq. (14) with respect to ReD . dΓ˙ dReD
1 + n + (n/2 + 4)C4 1 − 1/x Re−n−2 D −n/2+1 C1 Re dx × √ D 2x x(x − 1) dReD
2 −1 CC3 m−n/2+2 −n/2 × 1 + C1 ReD 1 − 1/x + ReD C2 √ dx (m − n/2 + 2) cosh−1 x 1 × − √ ReD 2 x(x − 1) dReD XReD dx 1− √ 2x x(x−1) dReD √ =W (1 + X 1 − 1/x )2 √ Z cosh−1 x 1 dx + − √ (15) Y ReD 2 x(x − 1) dReD
= −CRe−n−2 D
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Differentiating Eq. (12) with respect to ReD gives C1 Re−m−4 dx D = dReD C3 (RenD − C12 )3 × C14 C2 (m + 3) − nC12 (3C2 + K)RenD C1 C2 1−n/−1 ReD + (nK − pC2 )Re2n D + J × C13 C2 (3n − 2p) + C1 C2 (n + 2p)RenD Ren + K C1 − D C1 2
× C1 (2n − p) + (n + p)RenD where K = C1 C2 Rem+3 and D C1 (K + C2 ) J = XY −Z and ReD p=m+n+3 substituting the above equation in Eq. (15), we get 1 dΓ˙ = f (ReD , C1 , C2 , C3 ) C dReD
(16)
The minimum operational cost is found when Eq. (16) is equated to zero, this gives f (ReD , C1 , C2 , C3 ) = 0
(17)
It should be noted that m and n are known values for a given fin array and depend on ReD . Eq. (17) is a transcedental equation which can be solved for the optimum ReD , so that as a solution of this equation we can write ReDopt , = g(C1 , C2 , C3 )
(18)
This optimum value of (ReDopt ) if substituted in Eq. (11), will give optimum (ReLopt ) and if used in Eq. (14) will give the minimum cost of operation (Γ˙opt).
3. Results and discussion In the solution presented above C1 , C2 and C3 are recognized as the three dimensionless parameters influencing the cost of operation (Γ˙ ) for a pin-fin array (according to Eq. (9)) and the √ optimum fin geometry (see Eqs. (11) and (18)). C2 = 2 Cn /M is related to the fin material and the cooling fluid. For a typical application, using air as the cooling fluid (kf = 0.0265 W·m−1 ·K−1 ; Pr = 0.716) and Copper as the fin material (ks = 400 W·m−1 ·K−1 ), we get M = 130 (Eq. (3)), and C2 = 0.011 and 0.01455 for in-line and staggered arrangements, respectively. On the other hand if aluminium (ks = 237 W·m−1 ·K−1 ) is used as the fin material then M = 100, and C2 = 0.01437 and 0.0189 for inline and staggered arrangements, respectively. Therefore to
cover both Aluminium and Copper fin arrays, a range for C2 = 0.01–0.02 is chosen for the presented results. √ A parametric study for C1 = (π/2)(NV M/(G1 Cn )) is also performed to obtain a practical range. Considering variation of N and V from (3–30), M from (100–130), Sp /D and Sn /D from (1.25–4) and Cn from (0.27–0.9) yields a range of C1 = 11.5–682.6. Thus, a range for C1 = 10–1000 is chosen for the presented results. Parametric study for C3 = (B/2)(N/M 2 )(λP /λH )Cm × Cn G1 G2 indicated a wide range because of the large possible variations of B and λP /λH , the range for C3 = 10−12 –10−8 was selected for presenting results in graphical form. Fig. 2 shows the plot of dimensionless cost of operation (Γ˙ ∗ ) against ReD and ReL (Eq. (9)) for C1 = 1000, C2 = 0.015 and C3 = 10−11 . It indicates that there exist a point for minimum cost in the range of ReD and ReL shown. The minimum dimensionless cost was found as 4.32947 × 10−6 for ReD = 200.44 and ReL = 2159.83. Fig. 3 shows the plot of Eq. (14) which is the dimensionless cost obtained after the first step of optimization so that ReLopt (from Eq. (11)) has been substituted for ReL . The plot is shown for C1 = 1000, C2 = 0.015, and C3 = 10−11 and staggered arrangement. Two points of minimum A and B can be seen in the plot. Point A corresponds to ReD,A = 200.44 and point B corresponds to ReD,B = 3784.53. Using Eq. (11), ReLopt ,A = 2159.83 and ReLopt ,B = 40.056 and (L/D)A = 10.78 and (L/D)B = 0.0106. It is clear that L/D = 0.0106 is not a practical result although it corresponds to an overall minimum. Based on this, points similar to A which are local minimum in the range of L/D 1 were selected for determining ReDopt . Thus all the results presented below are subjected to the constraint that L/D is either of the order of 1 or greater than 1. Fig. 4 shows the plot of optimum Reynolds number (ReDopt ) against C3 for various values of C1 and C2 using Eq. (18). It can be seen that as C3 increases ReDopt decreases to low values which is reasonable, since low C3 implies small values of the fluid friction irreversibility coefficient as can be seen from the dimensionless variable B. In addition if (λP /λH ) is large, i.e., unit cost of pressure drop is more than unit cost of heat transfer, then also it is more desirable to make fins with smaller diameter (i.e., thin fins). The effect of C1 on ReDopt can also be seen from Fig. 4. As C1 increases (e.g., an array with more number of fins) the diameter of each fin should reduce for optimum cost of operation. The √ effect of C2 = 2 Cn /M is not as significant as C1 and C3 on ReDopt in the presented practical range of the values of C2 from 0.01–0.02.The small variation shows that as C2 increases (i.e., M = ks /kf /Pr1/6 becomes smaller) ReDopt also increases. The sudden variation in the results for ReDopt around ReD = 100 and ReD = 1000 for all results shown in Fig. 4 is due to different values of n in the different ranges of ReD , as shown in Table 1. Fig. 5 shows the variation of ReLopt with C1 , C2 and C3 . The effect of C3 on ReLopt is similar to the effect of C3
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253
Fig. 2. Plot of the dimensionless cost of operation (Γ˙ ∗ ) against ReD and ReL for a pin-fin array showing the point of minimum cost of operation.
Fig. 3. Plot of dimensionless cost of operation (Γ˙ ∗ ) and the corresponding L/D ratio against ReD showing two minimums A and B, and the range for which L/D is > 1.
on ReDopt . As C3 increases ReLopt decreases for all ranges of C1 and C2 considered in this study. However as C1 increases from 10–100, ReLopt increases and then decreases for C1 = 1000. Fig. 6 shows the plot of L/D ratio against C3 for different values of C1 and C2 . For C1 = 10, L/D is of the order of 1, since ReDopt and ReLopt are almost same. In
this case, for lower values of C2 , L/D even becomes lower than 1. For C1 = 100 and 1000, L/D is higher (varies in the range 5–40) with the variation of C3 from 10−12 –10−8 , since for these values ReDopt decreases but ReLopt increases. Fig. 7 shows the dimensionless optimum cost of opera∗ for different values of C , C and C . As C intion Γ˙opt 1 2 3 3
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Fig. 4. Plot of optimum Reynolds number (ReDopt ) against C3 for various values of C1 and C2 .
Fig. 5. Plot of optimum Reynolds number (ReLopt ) against C3 for various values of C1 and C2 .
creases the cost of operation also increases for all values of C1 and C2 considered, which is what is anticipated, because increase of C3 implies large fluid friction irreversibility coefficient B (refer to Eq. (A.8)) and small M (fins with smaller ks /kf ratio). The figure also shows that as C1 decreases (less no. of fins), the optimum cost of operation also decreases, reaching a minimum for large C1 . The variable C2 has less effect on the optimum cost for the range of values of C2 considered in the present study. The sudden variation in cost appearing in Fig. 7 is due to the sudden change in parameter n for the various ranges of ReD as shown in Table 1. It should be noted that for various ranges of the ReD appearing in the results, the parameter n is same for staggered and in-line fin arrangement (see Table 1). Thus
all the figures presented above are valid for both staggered and in-line arrangements. It should, however, be noted that ∗ would be different for the results for ReDopt , ReLopt and Γopt the two arrangements because C1 , C2 and C3 are different for the two cases as shown in the example, discussed below. Illustrative example: To explain the methodology presented in the present study, we consider the case of an aluminium 10 × 10, pin-fin array (ks = 237 W·m−2 ·K−1 ) which is to be optimized for usage in electronics cooling. The cooling fluid is air at an average temperature of 300 K (ρ = 1.1644 kg·m−3 ; ν = 1.6 × 10−5 m2 ·s−1 ; kf = 0.0265 W·m−2 ·K−1 ; Pr = 0.712) with the maximum flow velocity, Umax = 1 m·s−1 and a heat transfer rate of 0.5 W
S.Z. Shuja / Exergy, an International Journal 2 (2002) 248–258
Fig. 6. Plot of L/D against C3 for various values of C1 and C2 .
∗ ) against C for various values of C and C . Fig. 7. Plot of optimum dimensionless cost (Γ˙opt 3 1 2
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Table 2 Calculated values of hydraulic, thermal and geometric coefficients Cn (Table 1)
Cm (Table 1)
G1 (Eq. (A.9))
G2 (Eq. (A.9))
0.52 0.71
1.02951 1.74843
172.683 185.759
5.850 6.175
In-line Staggered
Table 3 Calculated values of thermoeconomic coefficients λ F M2
C= H Cn G 1
C2 = 2 MCn
λ C3 = B2 N2 λ P Cm Cn G1 G2 M H
126.145 100.355
0.014422 0.016852
3.6677 × 10−11 9.6572 × 10−11
G 1 Cn
8.158 × 10−4 5.554 × 10−4
In-line Staggered
√
C1 = π2 NV√M
Table 4 Final solution of optimum geometric parameters W
X
1.6669 × 10−7 1.4460 × 10−7
30.6145 25.3591
ReDopt Eq. (18) or Fig. 4 In-line Staggered
Y
Z
2.062 × 10−4 2.719 × 10−4
2.15 × 10−5 3.88 × 10−5
Eq. (7)
288 245
Table 5 Final solution of optimum geometric and cost values
In-line Staggered
x
ReLopt
Γ˙opt (ReLopt , ReDopt ) $·yr−1
Eq. (12)
Eq. (13) or Fig. 5
Eq. (14) or Fig. 7
1.27324 1.23243
2432.3 1710.6
0.6212 0.6820
for the fin array. The unit cost parameter associated with the heat transfer irreversibility, λH = 0.001$·K·J−1 ), and the cost ratio λP /λH = 0.1. The fin spacing ratio Sn /D = 1.65 and Sp /Sn = 1. Solution: The entropy parameter (F ) and the dimensionless parameter that accounts for the importance of fluid friction irreversibility relative to heat irreversibility (B) are q 2 Umax F = B 2 = 0.007325 W·K−1 νks T∞ B=
ρν 3 k
s T∞ 2 qB
= 1.3564 × 10−9
while the material parameter M is ks /kf M= = 100 Pr1/6 The other dimensionless parameters are calculated in Table 2. Now using Eq. (8) the solution for dimensionless parameters are in Table 3. Furthermore, the final solution can be tabulated in Tables 4 and 5. It can be seen from the above results that for similar thermoeconomic operating conditions, Dopt and Lopt for
Dopt (mm)
Lopt (mm)
4.608 3.920
38.92 27.37
the in-line arrangement of the fins is about 17% and 42%, respectively, more than the staggered configuration. In addition, although the cost of operation for in-line arrangement is less than that corresponding to the staggered arrangement, but the size and, therefore, corresponding material cost are more for the in-line arrangement.
4. Conclusions Optimum geometry of a pin fin array (with diameter and length as variables) is obtained by exergoeconomic analysis. Both in-line and staggered arrangements are studied. Practical ranges of the three solution parameters C1 , C2 and C3 defined through Eq. (8) are considered. Results are presented in analytical form for ReDopt , ReLopt and Γ˙opt through Eqs. (18), (13) and (14), respectively. Graphical solutions are also given for approximate handy calculations and fast interpretation. The parameter C3 representing fluid friction irreversibility coefficient B and unit cost ratio (λP /λH ) as well as the parameter C1 representing the number of fins and fin geometry, both affect the results such that the minimum cost increases as C3 increases and decreases as C1 increases. In addition, the optimum Reynolds number ReDopt decreases
S.Z. Shuja / Exergy, an International Journal 2 (2002) 248–258
with increase in C3 and C1 . C2 which has a practical range of 0.01–0.02 does not significantly affect the minimum cost of operation as well as ReDopt , although it is shown to have more affect on ReLopt . It is important to note that the presented results are more relevant where the operation is to be in the low ranges of Reynolds number, which is practical for cooling of electronic components, particularly in the mixed convection or purely natural convection regimes.
Acknowledgements The author acknowledge the support provided by King Fahd University of Petroleum and Minerals for this research project.
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D2 2 π NV kf mf L tanh(mf L) S˙gen = qB2 T∞ 4 L −1 π 2 + hw A − NV D 4 3 L(V − a)(S − D) CD NρUmax n (A.6) 2T∞ After non-dimensionalizing the diameter D as ReD and the length L as ReL , and assuming that hw = hf = h, the above equation can be rewritten as kf √ kf √ ReL π ˙ NV Nu tanh 2 Nu Sgen = F ReD 2 ks ks ReD −1 kf N + Nu G1 + F BReL ReD CD G2 ks 2 (A.7)
+
where Appendix A. Second-law analysis for a pin-fin array The irreversible losses of a fin array have been shown to consist mainly of those due to heat transfer and pressure drop. Thus the traditional irreversibility rate based objective function for a fin can be expressed as [13], qB θB mP ˙ S˙gen = 2 + ρT∞ T∞
(A.1)
In the above equation the term on the right-hand side indicate, the entropy generated due to heat transfer, and fluid friction. In this solution it has been assumed that the temperature difference, θB = TB − T∞ , is small compared with the absolute temperature, i.e., θB T∞ . According to the unidirectional heat conduction model for a fin array with constant cross-section area fins, the base stream temperature difference is given by [19], D2 π NV kf (mf L) tanh(mf L) θB = qB 4 L π + hw A − NV D 2 (A.2) 4 where mf =
F=
4hf ks D
B=
ρν 3 ks T∞ qB2
and the dimensionless geometric parameters are Sp Sn +1 G1 = (N − 1) Sn D π Sn × (V − a) + 1 − NV D 4 Sn G2 = (V − a) −1 D
(A.8)
(A.9)
In Eq. (A.7), the entropy generation is a summation of two terms which can be separated as S˙T and S˙P ; i.e., entropy generation due heat transfer and pressure drop, respectively, as kf √ π NV Nu S˙T = F ReD 2 ks −1 kf kf √ ReL Nu × tanh 2 + Nu G1 ks ReD ks (A.10) S˙P
qB2 Umax , 2 νks T∞
N = F B ReL ReD CD G2 2
(A.11)
and
A = (N − 1)Sp + D (V − a)Sn + D
(A.3)
Also, the mass flow rate is defined as m ˙ = ρUmax L(V − a)(Sn − D)
(A.4)
where Umax is the maximum average fluid velocity occurring at the minimum free flow area of the fin array. The pressure drop is defined using the drag coefficient, as [19] 1 2 ρU P = CD N (A.5) 2 max Using Eqs. (A.2), (A.4) and (A.5) in Eq. (A.1), we get
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