Arab J Sci Eng DOI 10.1007/s13369-014-1474-z
RESEARCH ARTICLE - MECHANICAL ENGINEERING
Simple Correlation to Evaluate Efficiency of Annular Elliptical Fin Circumscribing Circular Tube Hossain Nemati · Sina Samivand
Received: 28 June 2014 / Accepted: 29 October 2014 © King Fahd University of Petroleum and Minerals 2014
Abstract Efficiency of annular elliptical fin has been studied numerically. Results show that constant temperature lines do not remain circular in lower aspect ratio fins. So, it is expected that some common methods such as equivalent fin area or sector method do not have enough accuracy. For this reason, at first, a new simple correlation has been proposed to approximate annular circular fin efficiency based on longitudinal fin efficiency and then, comparing to the numerical results, it has been shown that the proposed formula is still applicable if one uses arithmetic mean of elliptical fin lengths and geometric mean of elliptical fin radiuses. Keywords Annular fins · Elliptical fin · Fin efficiency · Extended surface
List of symbols h Convection heat transfer coefficient (W/m2 K) First kind modified Bessel function of order n In k Fin thermal conductivity (W/mK) Second kind modified Kn Bessel function of order n L Fin length, r f − rb (m) L Arithmetic mean of major and minor fin lengths, L 1 +L 2 2 (m) Major fin lengths, (r1 − rb ) (m) L1 Major and minor fin lengths, (r L2 2 − rb ) (m) m N r r1 r2 rb rf Rf Rf1 Rf2 Rf s t T∞ Tb
H. Nemati (B) Marvdasht Branch, Department of Mechanics, Faculty of Mechanical Engineering, Islamic Azad University, Marvdasht, Iran e-mail:
[email protected] S. Samivand Aligoodarz Branch, Department of Mechanical Engineering, Islamic Azad University, Aligoodarz, Iran
Thermo-geometry parameters, kth (m−1 ) Normal vector Radial distance from tube center (m) Fin major radios (m) Fin minor radius (m) Tube radius (m) Circular fin radius (m) Normalized fin radius,r f /rb Major normalized radiuses, r1 /rb Minor normalized radiuses, r2 /rb Geometricmean of major and minor normalized radiuses, R f 1 R f 2 Fin efficiency parameter, Table 1 Fin semi-thickness (m) Ambient temperature (K) Fin base temperature (K)
Greek symbols γ Dimensionless group, m L R f /(R f − 1) η Fin efficiency φ Angular position from horizontal symmetric line (rad) ψ Fin efficiency parameter, Table 1 θ (T − T∞ )
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1 Introduction Fins or extended surfaces are used vastly in industry to increase the heat transfer area. The usefulness of fin employing is judged by fin efficiency. Fin efficiency is defined as the ratio of the actual heat transfer to the ideal heat transfer in which all fin area is in its base temperature. One of the most fins commonly used in process and waste heat recovery industries is annular fin [1]. Circular type of annular fin has been studied widely experimentally and numerically [2–5]. Even the performance of extended surface around elliptical tube or oval tube is studied [6,7]. However, the elliptical annular fin has not been studied yet. Pressure drop across elliptical fin is lesser than circular fin [8]. Moreover, since the heat transfer coefficient on the front side of the fin is more than both sides, using elliptical fin seems reasonable. Especially, when there is space restriction; elliptical fin can be a good candidate. With those aforementioned priorities, to the best knowledge of authors, no formula may be found in the literature to predict annular elliptical fin efficiency. Nagarani and Mayilsamy [8,9] performed some limited experiments to study the natural convection on specified elliptical fin geometry. Kunduand Das [10] proposed a semianalytical method similar to sector method to approximate elliptical fin efficiency. However, this method ends to a tedious series of Bessel function families. In this study, equivalent thermo-geometry parameters are proposed to be used in circular fin efficiency formula to estimate elliptical fin efficiency. This idea unifies the elliptical and circular fin efficiency correlation. However, the exact solution of circular fin efficiency is a combination of modified Bessel functions, which is very tedious. Several attempts have been made to approximate the exact solution with a simple correlation [11–14]; however, none of them has enough accuracy and simplicity. For this reason, at first step, a simple but accurate correlation has been proposed to approximate circular fin efficiency and at the next step, several numerical simulations have been performed to calculate elliptical fin efficiency in a vast range of thermo-geometry parameters. Finally, based on proposed circular fin efficiency formula and numerical results, equivalent parameters have been proposed to predict elliptical fin efficiency based on circular fin efficiency formula.
2 Formulation of the Problem An annular elliptical fin surrounding a circular tube is shown in Fig. 1. Fin thickness is 2t and tube radius is rb . The major and minor radiuses of ellipse are r1 and r2 , respectively. It is assumed that heat is exchanged with constant temperature ambient, T∞ only by convection and also, the convection
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Fig. 1 Annular elliptical fin surrounding a circular tube
coefficient, h, is constant. Based on the above assumptions, the steady state energy equation for constant fin thermal conductivity, k, in polar coordinate system may be written as: ∂ ∂r
∂θ 1 ∂ 2θ r + = m2θ ∂r r ∂φ 2
(1)
Due to symmetric form of geometry and heat transfer, only a quarter of fin is required to be considered. Boundary conditions are assumed as follows: • •
Outer periphery of the tube is assumed at constant temperature, Tb Heat transfer from symmetry lines and fin tip are zero.
In this regard, the boundary conditions may be present mathematically as (Fig. 2): θ = θb at r = rb (0 ≤ θ ≤ π/2)
(2a)
∂θ/∂φ = 0 at φ = 0 (rb ≤ r ≤ r1 )
(2b)
∂θ/∂φ = 0 at φ = π/2 (rb ≤ r ≤ r2 )
(2c)
∂θ/∂ N = 0 at 0 ≤ φ ≤ π/2 (r = r1r2 / (r1 sin(φ))2 + (r1 cos(φ))2 )
(2d)
In the preceding equations, θ is (T − T∞ ) , m 2 = kth and N is normal vector. Equation (1) is separable and can be solved by separation of variables method [15]. However, for elliptical fin, the forth boundary condition, Eq. (2d), is not separable. So, no analytical solution may be found for Eq. (1). Kundu and Das [10] proposed to divided the fin boundary to finite number of segments and approximate each segment as a circular arc to satisfy Eq. (2d) for each segment. It is somehow similar to sector method [13]. However, it is known that this method results in a series of Bessel function families. Even, with
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the existence of developed software, this series evaluation is elaborate and tedious.
the term cos(mψ Ls), the aforementioned modifications are not satisfactory. In this regard, the following approximation is proposed for circular fin efficiency:
3 Better Approximation to Circular Fin Efficiency
The exact solution of Eq. (1) in circular case in which ∂ 2 θ /∂φ 2 = 0 and r1 = r2 = r f is [16]: η=
which is a combination of Bessel function families. So, it is not a good base to approximate the elliptical fin efficiency. The famous approximation to the exact solution of Eq. (1) has been proposed by Schmidt [11] based on longitudinal fin efficiency. Some modifications have been proposed [12, 13] to increase the accuracy of this approximation by price of more complexity and restrictions. The general form of longitudinal fin-based approximation is: tanh(mψ L) cos(mψ Ls) mψ L
(4)
in which L = r f − rb is fin length. The other parameters are presented in Table 1. It is worthwhile to mention that longitudinal fin efficiency is: η = tanh(m L)/(m L)
tanh(mψ L) mψ L
ψ (6)
in which:
I1 (γ )K 1 (γ /R f ) − I1 (γ /R f )K 1 (γ ) 2 , m L(R f + 1) I1 (γ )K 0 (γ /R f ) − I0 (γ /R f )K 1 (γ ) (3)
η=
η=
ψ = 1 + 0.17912 ln R f
(7)
This formula is simple and accurate and in the limiting case, R f = 1, returns to longitudinal fin efficiency formula. Comparison of relative errors of all expressions is presented in Fig. 3 in which E is defined as difference between exact value and approximation of efficiency divided by exact value.
4 Approximation of Elliptical Fin Efficiency To calculate the elliptical fin efficiency, one hundred eighty models were simulated numerically by traditional software, Fluent. Those models cover a vest range of thermo-geometry parameters for them: 1 ≤ r1 /r2 ≤ 10, 1.25 ≤ r1 /rb ≤ 12.5 and 0.6 ≤ m ≤ 2,700. Applied boundary conditions were the same as Eq. (2). A very fine structured mesh was used
(5)
In the limiting case in which R f = 1, those approximations presented by Eq. (4) should be coincided with longitudinal fin efficiency formula, Eq. (5). But, due to the presence of
Fig. 2 Assumed fin geometry in polar coordinate system
Table 1 Parameters for longitudinal fin approximation (R f = r f /rb )
Fig. 3 Comparison between different circular fin efficiency approximations for R f = 2.5
s
ψ
Longitudinal fin
0
Schmidt [3]
0
Hong and Webb [4]
0.1
Perrotin and Clodic [5]
0.1
1 1 + 0.35 ln R f 1 + 0.35 ln R f
1.5− R f mL 12 ln R f 1 + 0.3 + 0.26R 0.3 − 0.3 f 2.5
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RMSE=0.009
Proposed Formula
0.8
0.6
0.4
0.2
0 0
0.2
0.4
0.6
0.8
1
Numerical Results Fig. 6 Comparison between numerical results and predicted values Table 2 Characteristics of sample fins r1 (mm)
r2 (mm)
rb (mm)
2t (mm)
k (W/mK)
30
20
14.7
0.4
202
85
20
14.7
0.4
202
100%
r2 r1
= 0.9 (approximately circular),
Fin Efficiency
95%
Fig. 4 temperature contours, a for b for rr21 = 0.3 (noncircular)
90% 85% 80% 75% 70% 0.1
0.3
0.5
0.7
0.9
1.1
1.3
m.L2 Equivalent method
Sector Method
Proposed formula
Numerical Soluon
Fig. 7 Comparison among different method,
Fig. 5 Presented formula and numerical results for two limiting cases, R f = 1.185 and R f = 11.85
to be ensured about numerical results. It was observed that for r1 /r2 near to unity, lines of constant temperature remain circular, approximately (Fig. 4a). However, by decreasing
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r2 r1
= 0.66
radiuses ratio, temperature contours will no longer be circular (Fig. 4b). So, it is expected that usual methods such as sector method would not be applicable. Since circular fin is a special case of elliptical fin, it is expected that with some modifications in main parameters, L and R f , Eq. (6) be still applicable. Several combinations were examined and finally, it was found out that one can use arithmetic mean of major and minor fin lengths instead of L in Eq. (6) and geometric mean of major and minor normalized radiuses instead of R f to have elliptical fin efficiency. So, the following formula will cover all longitudinal, circular and elliptical annular fins.
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η=
tanh(mψ L)
ψ (8)
mψ L
ψ = 1 + 0.17912 ln R f
(9)
in which: L1 + L2 (r1 − rb ) + (r2 − rb ) = 2 2 r1 r2 R f = R f1R f2 = rb rb
L=
(10) (11)
Comparison between presented formula and numerical results in two limiting cases, R f = 1.185 and R f = 11.85 is shown in Fig. 5. A good agreement can be observed. 100%
Ai ηi η= Ai
Fin Efficiency
90% 80% 70% 60% 50% 40% 30% 0.1
Also, comparison between numerical results and those obtained by Eq. (8), are depicted in Fig. 6. As it was claimed, common methods such as equivalent fin area method or sector method are not applicable for elliptical fin with low aspect ratio. To show this, efficiency of two different elliptical fins are evaluated by different methods and the results are compared with the proposed formula, Eq. (7). The characteristics of both fins are indicated in Table 2. In the first one, rr21 = 0.66 while in the next one, rr21 = 0.23. In the equivalent fin area method, it is assumed that the efficiency of the fin is equal with the efficiency of a circular fin with the same area. Although this method is simple, it is inaccurate. On the other hand, in the sector method, the fin area is divided to some finite number of circular sectors. So, the efficiency may be approximated as:
0.3
0.5
0.7
0.9
1.1
1.3
m.L 2 Equivalent method
Sector Method
Proposed formula
Numerical Soluon
Fig. 8 Comparison among different method,
r2 r1
= 0.23
(12)
in which Ai , ηi is area and efficiency of ith sector. Results are depicted in Figs. 7 and 8. According to Fig. 7, since the fin shape is nearly circular, all methods predict efficiency well, however, as aspect ratio decreases as shown in Fig. 8, the equivalent fin area method no longer can predict the fin efficiency correctly. Moreover, sector method also loses its accuracy. The reason may be found in temperature contours. As it can be observed in Fig. 9, the predicted temperate contours are basically different from the real one in fins with lower aspect ratio. So, it seems reasonable not to expect that sector method can predict the fin efficiency correctly.
Fig. 9 Temperature contours: real (top) and sector method (bottom)
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5 Conclusions The efficiency of elliptical annular fin was studied at the present work. Since no analytical solution may be found to calculate fin efficiency, numerical method was employed for a vast range of thermo-geometric parameters. At the first stage, a new simple but accurate correlation was proposed to predict circular annular fin efficiency. In the next step, based on numerical results, it was shown that the proposed formula for annular circular fin is still applicable if one uses arithmetic mean of elliptical fin lengths and geometric mean of elliptical fin radiuses. So, the presented formula covers all longitudinal, circular and elliptical annular fins. Finally, the results were compared with equivalent fin area method and sector method for a typical elliptical fin. It was shown that although all methods in relatively circular fins predict fin efficiency well, their accuracies are not reliable in lower aspect ratio.
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