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Quick estimate of the heat transfer characteristics of annular fins of hyperbolic profile with the power series method
Applied Thermal Engineering 25 (2005) 623–634 www.elsevier.com/locate/apthermeng
Quick estimate of the heat transfer characteristics of annular fins of hyperbolic profile with the power series method Inmaculada Arauzo a, Antonio Campo a
b,*
, Cristo´bal Corte´s
a
Dpto. de Ingenierı´a Meca´nica, Universidad de Zaragoza, C/Marı´a de Luna 3, Zaragoza 50018, Spain b Department of Mechanical Engineering, The University of Vermont, Burlington, VT 05405, USA Received 15 May 2003; accepted 7 May 2004 Available online 18 September 2004
Abstract This technical paper addresses an elementary analytic procedure for the approximate solution of the quasi-one-dimensional heat conduction equation (a generalized Bessel equation) that governs the temperature variation in annular fins of hyperbolic profile. This fin shape is of remarkable importance because its heat transfer performance is close to that of the annular fin of convex parabolic profile, the so-called optimal annular fin that is capable of delivering maximum heat transfer for a given volume of material [Zeitschrift des Vereines Deutscher Ingenieure 70 (1926) 885]. The salient feature of the analytic procedure developed here is that for realistic combinations of the two parameters: the enlarged Biot number and the normalized radii ratio, the truncated power series solutions embracing a moderate number of terms yields unprecedented results of excellent quality. The analytic results are conveniently presented in terms of the two primary quantities of interest in thermal design applications, namely the heat transfer rates and the tip temperature. Ó 2004 Elsevier Ltd. All rights reserved.
*
Corresponding author. Tel.: +1 802 656 3320; fax: +1 802 656 1929. E-mail address: [email protected] (A. Campo).
1359-4311/$ - see front matter Ó 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.applthermaleng.2004.05.019
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I. Arauzo et al. / Applied Thermal Engineering 25 (2005) 623–634
Nomenclature c h Im k M2 Q Qideal r r1 r2 R Ro T Tb Tn Tt T1 u m d1 d2 u gdif gint h n
normalized radii ratio, r1/r2 convection heat transfer coefficient modified Bessel function of first kind and order m thermal conductivity enlarged Biot number, h r32 =ðk d1 r1 Þ heat transfer rate ideal heat transfer rate radial coordinate inner radius outer radius dimensionless r, r/r2 center of the power series temperature base temperature approximate temperature calculated by truncated power series with n terms tip temperature fluid temperature fin parameter, Eq. (7b) fin parameter, Eq. (7c) inner semi-thickness outer semi-thickness fin efficiency, Q/Qideal fin efficiency by way of differentiation fin efficiency by way of integration dimensionless T, (TT1)/(TbT1) fin parameter, Eq. (7d)
1. Introduction A common procedure for augmenting the transfer of heat from round tubes to adjacent fluid streams deals with the attachment of arrays of annular fins to the outer surface of the tubes for heat transfer equipment [1,2]. Typical applications of annular finned are found in air-cooled engines of motorcycles and automobiles, in gas–gas or gas–liquid heat exchangers for process industries, in condensers and evaporators for refrigeration cycles, etc. Back in 1926, Schmidt [3] discovered that for a given volume of material, the annular fin of convex parabolic profile was able to supply maximum heat transfer from the surface of a round tube to a surrounding fluid. As expected, this particular fin was appropriately named thereafter the optimal annular fin [4]. A drawback posed by the convex parabolic profile is its sharp tip, because this configuration may be difficult to manufacture and to maintain in a plant environment.
I. Arauzo et al. / Applied Thermal Engineering 25 (2005) 623–634
625
Among the existing family of annular fins possessing tapered cross sections, it is widely recognized that the annular fin of hyperbolic profile is the foremost fin shape candidate for practical applications [1,2]. From an optimization standpoint, the annular fin of hyperbolic profile closely resembles the optimal annular fin of convex parabolic profile. As far as the modeling is concerned, the temperature change along an annular fin of hyperbolic profile is governed by a quasi-onedimensional heat equation, the so-called generalized Bessel equation. Despite that this equation admits an analytical solution for combinations of the enlarged Biot number M2 and the normalized radii ratio c, the evaluation of local temperatures and heat transfer rates with modified Bessel functions of first kind and fractional order is complicated and time-consuming. To avoid these obstacles, the present technical paper addresses the power series method as an alternate computational procedure for solving the governing quasi-one-dimensional heat equation in approximate manner. Due to its inherent simplicity, the power series method may be attractive to thermal design engineers and also to instructors of graduate courses on heat transfer. Factors influencing the structure of the power series solutions and their exactness will be discussed at length.
2. Formulation Fig. 1 depicts the path of two symmetric hyperbolae y(r) = d1(r1/r) forming an annular fin of hyperbolic profile that features four dimensions: the inner radius r1, the inner semi-thickness d1, the outer radius r2 and the outer semi-thickness d2. The temperature variation along the annular fin of hyperbolic profile obeys the dimensionless fin equation [4]: d2 h M 2 R3 h ¼ 0 dR2 subject to the boundary conditions of prescribed temperature at the fin base R2
h ¼ 1;
R¼c
ð1Þ
ð2aÞ
and negligible heat loss at the fin tip dh ¼ 0; R ¼ 1 ð2bÞ dR The dimensionless variables for the temperature h and the radial coordinate R used in Eqs. (1) and (2) are T T1 r h¼ ; R¼ ð3Þ r2 Tb T1 The two parameters that surface up in the formulation are the enlarged Biot number M 2 ¼ hr32 =kd1 r1 in Eq. (1) and the normalized radii ratio 0 < c = r1/r2 6 1 in Eq. (2a). The heat transfer rate Q from a fin to a neighboring fluid is customarily computed indirectly with the fin efficiency g¼
Q Qideal
ð4Þ
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I. Arauzo et al. / Applied Thermal Engineering 25 (2005) 623–634
Fig. 1. Sketch of the annular fin of hyperbolic profile.
in two equivalent ways: (a) utilizing the derivative of h(R) at the fin base dh 2 dR R¼c g¼ 2 M ð1 c2 Þ or (b) employing the integral of h(R) over the fin length R1 2 c hR dR g¼ 1 c2
ð5aÞ
ð5bÞ
3. Exact analytic solution With a proper transformation, Eq. (1) falls under the category of a generalized Bessel type equation [5]. Accordingly, the exact solution of Eq. (1) united to Eqs. (2a) and (2b) delivers the exact dimensionless temperature distribution [4]: 3 2 rffiffiffi I 1=3 2 MR3=2 I 2=3 2 M I 1=3 2 MR3=2 I 2=3 2 M 7 R6 3 3 3 3 7 6 ð6Þ hðrÞ ¼ 5 2 3=2 2 2 3=2 2 c4 I 1=3 Mc I 2=3 M I 1=3 Mc I 2=3 M 3 3 3 3
I. Arauzo et al. / Applied Thermal Engineering 25 (2005) 623–634
and the exact fin efficiency: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 2cð1 cÞ I 2=3 ðuÞI 2=3 ðvÞ I 2=3 ðuÞI 2=3 ðvÞ g¼ nð1 c2 Þ I 2=3 ðuÞI 1=3 ðvÞ I 2=3 ðuÞI 1=3 ðvÞ where the following fin parameters: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 lnð1=cÞ u¼ n 3 ð1 cÞ3 2 v¼ n 3
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2c3 lnð1=cÞ ð1 cÞ3
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 ð1 cÞ n¼M 2 lnð1=cÞ
627
ð7aÞ
ð7bÞ
ð7cÞ
ð7dÞ
are introduced for the sake of compactness. In the tandem of Eqs. (6) and (7), Im stands for the modified Bessel functions of first kind and fractional order m = 1/3, 1/3, 2/3 and 2/3. It is evident that the numerical evaluation of local temperatures and fin efficiencies in Eqs. (6) and/or (7) is a laborious task even with the help of computer-aided tools, such as Mathematica, Maple, Matlab or others. From a perspective of thermal design, it should be realized that the fin efficiency diagram representative of the double-valued function g = f(n, c) is the only information thermal design engineers have at their disposal for the estimation of actual heat transfer rates from non-straight fins immersed in fluids. In this regard, it is surprising to see that in the fin efficiency diagram for annular fins of hyperbolic profile in [4] the family of curves g vs. n parameterized by c contains only three curves that correspond to the three normalized radii ratios: c = 1, 1/2 and 1/4. Predictably, for values of c different than those, interpolation between the readings of the three g-curves, is burdensome and somewhat inaccurate.
4. Approximate power series solution The power series method is a standard technique for solving linear ordinary differential equations with variable coefficients of the general type [6]: d2 y dy þ pðxÞ þ qðxÞy ¼ 0 ð8Þ 2 dx dx with various levels of approximation. In this equation, the variable coefficients p(x) and q(x) are most likely polynomials represented by power series. An approximate solution of Eq. (8) is viable in terms of a power series of the following form yðxÞ ¼
1 X i¼0
ai ðx x0 Þi ¼ a0 þ a1 ðx x0 Þ þ a2 ðx x0 Þ2 þ
ð9Þ
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I. Arauzo et al. / Applied Thermal Engineering 25 (2005) 623–634
where a0, a1, a2, . . . are the coefficients and x0 is the center of the power series. The theory specialized on the existence, uniqueness, and convergence of power series solutions may be found in Ince [7]. Since p(x) = 0 and q(x) = M2R is a monomial in Eq. (1), the power series method is appealing for solving this particular equation. Accordingly, the following summation: 1 X 2 m i ai ðR 1Þ ð10Þ hðRÞ ¼ a0 þ a1 ðR 1Þ þ a2 ðR 1Þ þ þ am ðR 1Þ ¼ i¼0
seems to be adequate to accomplish this task. For convenience, the fin tip, R = 1, is taken as the center of the power series. As a result, Eq. (10) reveals that the first coefficient a0 delivers the dimensionless tip temperature ht = h(1) = a0 right away. After calculating the second derivative of h(R) from Eq. (10), substituting the expression in Eq. (1) and carrying out the algebraic manipulations, we get the following system of two algebraic equations for the collection of coefficients, ai: i¼0:
2a2 M 2 a0 ¼ 0
i ¼ 1; . . . 1 :
ði þ 2Þði þ 1Þaiþ2 M 2 ðai þ ai1 Þ ¼ 0
ð11aÞ ð11bÞ
First, the boundary condition of negligible heat loss at the fin tip given by Eq. (2b) nullifies the second coefficient a1 of Eq. (10): a1 ¼ 0
ð12Þ
Second, the boundary condition of prescribed temperature at the fin base in Eq. (2a) brings forward an additional algebraic equation for the remaining coefficients, ai (i51): / X i ai ð1 cÞ ¼ 1 ð13Þ i¼0
Conceptually, the exact calculation of the general term in Eq. (10) would lead us to the exact solution having combinations of modified Bessel functions as seen in Eq. (6). However, we are trying to avoid this tortuous route in this work. To fulfill our definitive goal, we begin with the truncation of the power series by taking a finite number of terms from the sequence of Eqs. (10)–(13). In this methodical way, approximate expressions for the coefficients may be quickly deduced: (1) the coefficients ai are represented in terms of the first coefficient a0 (the dimensionless tip temperature) and M2 (the enlarged Biot number) and (2) the first coefficient a0 (the dimensionless tip temperature) depends on M2 (the enlarged Biot number) and the normalized radii ratio c. Unfortunately, under the premises of this generic format it is not possible to assess the convergence of the truncated power series. In fact, the convergence patterns of the truncated power series may vary from case to case depending on the values assigned to c and M2. For the examination of individual case studies, the values of M2 and c are determined from the specification of the geometrical, hydrodynamic and thermal quantities. In this sense, it may be demonstrated later that
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Table 1 General expressions from the second to the tenth coefficient of the power series a2 ¼
M2 a0 2!
M2 a0 3! 2 M þ1 a6 ¼ M 4 a0 6! 2 4M þ 7 a9 ¼ 4M 6 a0 9! a3 ¼
4M 4 a0 5! 2 M þ 28 a8 ¼ M 6 a0 8! a5 ¼
a4 ¼
M4 a0 4!
9M 6 a0 7! 2 M þ 100 ¼ M 8 a0 10!
a7 ¼ a10
the number of terms that must be retained to guarantee adequate levels of accuracy for h(R) grows with higher values of M2 and smaller values of c. After applying first Eq. (11) for i = 0, . . . , 8, the ensuing expressions for the coefficients a2, . . . , a10 expressed in terms of a0 and M2 are itemized in Table 1. Second, the leading coefficient a0 may be then calculated with Eq. (13). Next, in order to compute the fin efficiencies within the platform of the power series method, we substituted the approximate temperature distributions, h(R), as given by Eqs. (10)–(13) into the pair of Eqs. (5a) and (5b). Specifically, for a power series with m terms, the respective expressions for g are given by gdif
P 2 mi¼2 i ai ðc 1Þi2 ¼ M 2 ð1 þ cÞ
ð14Þ
and 2 gint ¼
Pm
i i¼0 ai ðc 1Þ
ð1 þ cÞ
c1 iþ2
1 þ iþ1
ð15Þ
Here, we called gdif the fin efficiency calculated by differentiation and gint the fin efficiency calculated by integration. As may be confirmed later, the above approximate g-relations are evaluated easily with the substitution of a finite number of coefficients ai presented in Table 1, in conjunction with the leading coefficient a0 as given by Eq. (13). It should be pointed out that by virtue of the two calculation options available for g, the integration approach via Eq. (15) yields better results than the differentiation approach via Eq. (14). The truncated power series of Eq. (10) is now complete and may be used with confidence for the computation of the temperature distributions (including tip temperatures) and fin efficiencies of any annular fin of hyperbolic profile. As an illustrative example, we report in the next section two sample calculations that convey the mechanics of the power series method retaining 5 and 10 terms for a pre-selected combination of c and M2. Incidentally, in the event that more than 10 coefficients ai need to be employed in the power series, any symbolic computer code or even a spreadsheet may facilitate the required
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I. Arauzo et al. / Applied Thermal Engineering 25 (2005) 623–634
numerical calculations, since it is only a question of solving a system of linear algebraic equations.
5. Practical example Observation of the fin efficiency diagram in Ref. [4] reveals that the modified Biot number M2 in the abscissa goes from 0 to 10. In addition, the normalized radii ratio c being a parameter extends from a small value of c = 1/4 (long annular fin of hyperbolic profile) to the limiting large value of c = 1 (straight fin of rectangular profile). The format of the fin efficiency diagram insinuates that the largest outer-to-inner radii ratio normally used for annular finned tubes, stays around 4. In view of this variability, the choice of c = 1/4 identifying the bottom curve in the fin efficiency diagram in [4] seems to be a logical choice because it represents the worst possible scenario in real applications of heat exchange devices [1,2]. For the specific fin size dictated by c = 1/4, we selected an intermediate value of the modified Biot number, say M2 = 6.572, so that the annular fin of hyperbolic profile operates with a fin efficiency g of a moderate magnitude 0.520. Now, the implementation of the power series method is undertaken systematically with the goal at calculating the temperature distribution along the fin and the heat transfer rate from the fin to the fluid. Let us begin with a 5-term power series as delineated in the preceding section. Doing the algebra, the approximate dimensionless temperature distribution h5(R) results in h5 ðRÞ ¼ 0:38255 þ 1:25710ðR 1Þ2 þ 0:41902ðR 1Þ3 þ 0:68845ðR 1Þ4 þ 0:55076ðR 1Þ5 ð16Þ Doubling the number of terms in a second try, the approximate dimensionless temperature distribution h10(R) based on a 10-term power series turns out to be h10 ðRÞ ¼ 0:37329 þ 1:2266ðR 1Þ2 þ 0:40888ðR 1Þ3 þ 0:67178ðR 1Þ4 þ 0:53743ðR 1Þ5 þ þ0:23674ðR 1Þ6 þ 0:18921ðR 1Þ7 þ 0:09085ðR 1Þ8 þ 0:03888ðR 1Þ9 þ 0:02045ðR 1Þ10
ð17Þ
Fig. 2 displays the closeness of the two approximate temperature distributions h5(R) and h10(R), as compared against the exact temperature distribution hexact(R) coming from Eq. (6). Herewith, it may be confirmed that both the 5- and the 10-term series handle the temperature variation along the annular fin of hyperbolic profile admirably. The largest error in the predicted temperature occurs at the fin tip, as could have been expected. The resulting two approximate tip temperatures are ht = h5(1) = 0.38255 and ht = h10(1) = 0.37329. Using as a reference the exact tip temperature hexact(1) in Eq. (6), h10(1) overlaps with hexact(1) whereas h5(1) exhibits an error of order 2.5% with respect to hexact(1). To guarantee accuracy, the exact temperatures and the exact fin efficiencies in Eqs. (6) and (7) were evaluated numerically with the symbolic code Maple [8].
I. Arauzo et al. / Applied Thermal Engineering 25 (2005) 623–634
631
θ 1 θ θ _5 term θ _10 term
0.9
0.8
0.7
0.6
0.5
0.4
0.3 0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
R
Fig. 2. Approximate temperature distributions for M2 = 6.572 and c = 1/4, using a 5- and a 10-term power series.
6. General cases To assess the convergence of the 10-term power series temperature distribution, h10(R), the crucial test involves the approximate tip temperatures. Hence, h10(1) in Eq. (17) along with the exact tip temperatures hexact(1) in Eq. (6) are compared for all possible combinations of the fin parameter n and the normalized radii ratio c. The outcome of the comparison is displayed in Fig. 3. It is observable in the figure that the agreement is excellent, the only subtle deviations take place in the limiting condition related to large n and small c, i.e., n = 3 and c = 0.1. Further, it should be recognized that the uppermost curve for c = 1 is actually the limiting situation of the annular fin of hyperbolic profile symbolizing a straight fin of rectangular profile (r2 r1)/r2 ! 0. Herewith, the tip temperature is calculated by the ratio: hð1Þ ¼
1 pffiffiffi coshð 2nÞ
ð18Þ
The exact fin efficiency curves are obtained from the evaluation of Eq. (7) for all combinations of c and M2. The corresponding numbers are compared against the approximate fin efficiencies computed by means of Eq. (15) united to the 10-term temperature distribution h10(R) in Eq. (17). It should be mentioned that the first curve for c = 1 is actually the limiting case of a straight fin of uniform profile, (r2 r1)/r2 ! 0 whose fin efficiency is calculated as: pffiffiffi tanhð 2nÞ pffiffiffi g¼ ð19Þ 2n
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I. Arauzo et al. / Applied Thermal Engineering 25 (2005) 623–634 θ(1)
1 0.9 0.8 0.7 0.6 0.5 0.4 c=0.5 c=0.7 c=1
0.3 0.2
c=0.1
c=0.2 c=0.3
0.1 0 0
0.5
1
1.5
2
2.5
3
ξ
Fig. 3. Approximate tip temperatures using a 10-term power series (black dots) and the exact tip temperatures (solid lines).
The agreement between the exact and the approximate estimates of g is perfect for all radii ratios contained in the large interval 0.2 6 c 6 1 and values of the fin parameter n that lie in the interval 0 6 n 6 2. These combinations of c and n cover an ample gamma of fin efficiencies that extend between 0.25 and 1. For the fin holding a radii ratio c = 0.1 the last curve indicates that the agreement is also perfect but it is applicable up to n = 1.5 inclusive. To save journal space, we decided not to include the differential-based calculations of g utilizing a 10-term power series, i.e. Eqs. (14) and (17). From the framework of thermal performance per se, it is known that regardless of the cross sectional shape, the addition of fins is beneficial when they deliver efficiencies with magnitudes near one [1,2]. In other words, this statement is indicative of efficiency numbers that lie in the upper left region of Fig. 4. For completeness, it should be added that the 5-term series in Eq. (16) gives reasonably good results for the same values of c coupled with values of n up to 1 and even up to 1.5 for c P 0.5. Contrary to the trends manifested in Fig. 4, the fin efficiencies based on 5-term series tend to be slightly overestimated. Owing to the detailed characterization of dimensionless radii ratios c, it may be categorically stated that the complete fin efficiency diagram of Fig. 4 containing six curves supersedes the reduced fin efficiency diagram of Ref. [4] that contains only three curves.
I. Arauzo et al. / Applied Thermal Engineering 25 (2005) 623–634 η
1 0.9 0.8 0.7
c=0.1
η_int c=0.1
c=0.2
η_int c=0.2
c=0.3
η_int c=0.3
c=0.5
η_int c=0.5
c=0.7
η_int c=0.7
633
c=1
0.6 0.5 0.4 0.3
c = 1.0 c = 0.7 c = 0.5 c = 0.3 c = 0.2 c = 0.1
0.2 0.1 0 0
0.5
1
1.5
2
2.5
3
ξ
Fig. 4. Approximate fin efficiencies using a 10-term power series (black dots) and the exact fin efficiencies (solid lines).
7. Conclusions The major conclusion that may drawn from this technical paper is that a 10-term power series is splendid for predicting flawlessly the temperature distributions and the companion heat transfer rates of annular fins of hyperbolic profile of practical significance in heat transfer engineering. Compared to the exact temperature distribution and the exact heat transfer rates evaluated with intricate Bessel functions of fractional order, it was realized that the 10-term power series solution is versatile, offers considerable economy and more importantly supplies accurate results. The analytic expressions for the participating series coefficients are compact, permitting the study of a wide range of realistic combinations of the two controlling parameters: the normalized radii ratio c and the enlarged Biot number M2. In principle, the power series method being the precursor of the venerable Bessel functions may be extended with ease to a general class of straight or annular fins of tapered profile in which the common straight or annular fin of rectangular profile are just particular cases. Acknowledgments The authors are indebted to Richard D. Kirkham, College of Engineering, Idaho State University, Pocatello, Idaho for his help during the early stages of the calculations with the power series method.
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I. Arauzo et al. / Applied Thermal Engineering 25 (2005) 623–634
References [1] A.D. Kraus, A. Aziz, J.R. Welty, Extended surface heat transfer, Wiley, New York, 2000. [2] R.L. Webb, Principles of enhanced heat transfer, Wiley, New York, 1994. [3] E. Schmidt, Die wa¨rmeu¨bertragung durch rippen, Zeitschrift des Vereines Deutscher Ingenieure 70 (1926) 885–889, pp. 947–951. [4] P.J. Schneider, Conduction heat transfer, Addison-Wesley, Reading, MA, 1955. [5] V. Arpaci, Conduction heat transfer, Addison-Wesley, Reading, MA, 1966. [6] W.E. Boyce, R.C. DiPrima, Elementary differential equations and boundary value problems, Wiley, New York, 1965. [7] E.L. Ince, Ordinary differential equations, Dover, New York, 1956. [8] E. Kamerich, A guide to maple, Springer-Verlag, New York, 1998.
Quick estimate of the heat transfer characteristics of annular fins of hyperbolic profile with the power series method Inmaculada Arauzo a, Antonio Campo a
b,*
, Cristo´bal Corte´s
a
Dpto. de Ingenierı´a Meca´nica, Universidad de Zaragoza, C/Marı´a de Luna 3, Zaragoza 50018, Spain b Department of Mechanical Engineering, The University of Vermont, Burlington, VT 05405, USA Received 15 May 2003; accepted 7 May 2004 Available online 18 September 2004
Abstract This technical paper addresses an elementary analytic procedure for the approximate solution of the quasi-one-dimensional heat conduction equation (a generalized Bessel equation) that governs the temperature variation in annular fins of hyperbolic profile. This fin shape is of remarkable importance because its heat transfer performance is close to that of the annular fin of convex parabolic profile, the so-called optimal annular fin that is capable of delivering maximum heat transfer for a given volume of material [Zeitschrift des Vereines Deutscher Ingenieure 70 (1926) 885]. The salient feature of the analytic procedure developed here is that for realistic combinations of the two parameters: the enlarged Biot number and the normalized radii ratio, the truncated power series solutions embracing a moderate number of terms yields unprecedented results of excellent quality. The analytic results are conveniently presented in terms of the two primary quantities of interest in thermal design applications, namely the heat transfer rates and the tip temperature. Ó 2004 Elsevier Ltd. All rights reserved.
*
Corresponding author. Tel.: +1 802 656 3320; fax: +1 802 656 1929. E-mail address: [email protected] (A. Campo).
1359-4311/$ - see front matter Ó 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.applthermaleng.2004.05.019
624
I. Arauzo et al. / Applied Thermal Engineering 25 (2005) 623–634
Nomenclature c h Im k M2 Q Qideal r r1 r2 R Ro T Tb Tn Tt T1 u m d1 d2 u gdif gint h n
normalized radii ratio, r1/r2 convection heat transfer coefficient modified Bessel function of first kind and order m thermal conductivity enlarged Biot number, h r32 =ðk d1 r1 Þ heat transfer rate ideal heat transfer rate radial coordinate inner radius outer radius dimensionless r, r/r2 center of the power series temperature base temperature approximate temperature calculated by truncated power series with n terms tip temperature fluid temperature fin parameter, Eq. (7b) fin parameter, Eq. (7c) inner semi-thickness outer semi-thickness fin efficiency, Q/Qideal fin efficiency by way of differentiation fin efficiency by way of integration dimensionless T, (TT1)/(TbT1) fin parameter, Eq. (7d)
1. Introduction A common procedure for augmenting the transfer of heat from round tubes to adjacent fluid streams deals with the attachment of arrays of annular fins to the outer surface of the tubes for heat transfer equipment [1,2]. Typical applications of annular finned are found in air-cooled engines of motorcycles and automobiles, in gas–gas or gas–liquid heat exchangers for process industries, in condensers and evaporators for refrigeration cycles, etc. Back in 1926, Schmidt [3] discovered that for a given volume of material, the annular fin of convex parabolic profile was able to supply maximum heat transfer from the surface of a round tube to a surrounding fluid. As expected, this particular fin was appropriately named thereafter the optimal annular fin [4]. A drawback posed by the convex parabolic profile is its sharp tip, because this configuration may be difficult to manufacture and to maintain in a plant environment.
I. Arauzo et al. / Applied Thermal Engineering 25 (2005) 623–634
625
Among the existing family of annular fins possessing tapered cross sections, it is widely recognized that the annular fin of hyperbolic profile is the foremost fin shape candidate for practical applications [1,2]. From an optimization standpoint, the annular fin of hyperbolic profile closely resembles the optimal annular fin of convex parabolic profile. As far as the modeling is concerned, the temperature change along an annular fin of hyperbolic profile is governed by a quasi-onedimensional heat equation, the so-called generalized Bessel equation. Despite that this equation admits an analytical solution for combinations of the enlarged Biot number M2 and the normalized radii ratio c, the evaluation of local temperatures and heat transfer rates with modified Bessel functions of first kind and fractional order is complicated and time-consuming. To avoid these obstacles, the present technical paper addresses the power series method as an alternate computational procedure for solving the governing quasi-one-dimensional heat equation in approximate manner. Due to its inherent simplicity, the power series method may be attractive to thermal design engineers and also to instructors of graduate courses on heat transfer. Factors influencing the structure of the power series solutions and their exactness will be discussed at length.
2. Formulation Fig. 1 depicts the path of two symmetric hyperbolae y(r) = d1(r1/r) forming an annular fin of hyperbolic profile that features four dimensions: the inner radius r1, the inner semi-thickness d1, the outer radius r2 and the outer semi-thickness d2. The temperature variation along the annular fin of hyperbolic profile obeys the dimensionless fin equation [4]: d2 h M 2 R3 h ¼ 0 dR2 subject to the boundary conditions of prescribed temperature at the fin base R2
h ¼ 1;
R¼c
ð1Þ
ð2aÞ
and negligible heat loss at the fin tip dh ¼ 0; R ¼ 1 ð2bÞ dR The dimensionless variables for the temperature h and the radial coordinate R used in Eqs. (1) and (2) are T T1 r h¼ ; R¼ ð3Þ r2 Tb T1 The two parameters that surface up in the formulation are the enlarged Biot number M 2 ¼ hr32 =kd1 r1 in Eq. (1) and the normalized radii ratio 0 < c = r1/r2 6 1 in Eq. (2a). The heat transfer rate Q from a fin to a neighboring fluid is customarily computed indirectly with the fin efficiency g¼
Q Qideal
ð4Þ
626
I. Arauzo et al. / Applied Thermal Engineering 25 (2005) 623–634
Fig. 1. Sketch of the annular fin of hyperbolic profile.
in two equivalent ways: (a) utilizing the derivative of h(R) at the fin base dh 2 dR R¼c g¼ 2 M ð1 c2 Þ or (b) employing the integral of h(R) over the fin length R1 2 c hR dR g¼ 1 c2
ð5aÞ
ð5bÞ
3. Exact analytic solution With a proper transformation, Eq. (1) falls under the category of a generalized Bessel type equation [5]. Accordingly, the exact solution of Eq. (1) united to Eqs. (2a) and (2b) delivers the exact dimensionless temperature distribution [4]: 3 2 rffiffiffi I 1=3 2 MR3=2 I 2=3 2 M I 1=3 2 MR3=2 I 2=3 2 M 7 R6 3 3 3 3 7 6 ð6Þ hðrÞ ¼ 5 2 3=2 2 2 3=2 2 c4 I 1=3 Mc I 2=3 M I 1=3 Mc I 2=3 M 3 3 3 3
I. Arauzo et al. / Applied Thermal Engineering 25 (2005) 623–634
and the exact fin efficiency: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 2cð1 cÞ I 2=3 ðuÞI 2=3 ðvÞ I 2=3 ðuÞI 2=3 ðvÞ g¼ nð1 c2 Þ I 2=3 ðuÞI 1=3 ðvÞ I 2=3 ðuÞI 1=3 ðvÞ where the following fin parameters: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 lnð1=cÞ u¼ n 3 ð1 cÞ3 2 v¼ n 3
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2c3 lnð1=cÞ ð1 cÞ3
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 ð1 cÞ n¼M 2 lnð1=cÞ
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ð7aÞ
ð7bÞ
ð7cÞ
ð7dÞ
are introduced for the sake of compactness. In the tandem of Eqs. (6) and (7), Im stands for the modified Bessel functions of first kind and fractional order m = 1/3, 1/3, 2/3 and 2/3. It is evident that the numerical evaluation of local temperatures and fin efficiencies in Eqs. (6) and/or (7) is a laborious task even with the help of computer-aided tools, such as Mathematica, Maple, Matlab or others. From a perspective of thermal design, it should be realized that the fin efficiency diagram representative of the double-valued function g = f(n, c) is the only information thermal design engineers have at their disposal for the estimation of actual heat transfer rates from non-straight fins immersed in fluids. In this regard, it is surprising to see that in the fin efficiency diagram for annular fins of hyperbolic profile in [4] the family of curves g vs. n parameterized by c contains only three curves that correspond to the three normalized radii ratios: c = 1, 1/2 and 1/4. Predictably, for values of c different than those, interpolation between the readings of the three g-curves, is burdensome and somewhat inaccurate.
4. Approximate power series solution The power series method is a standard technique for solving linear ordinary differential equations with variable coefficients of the general type [6]: d2 y dy þ pðxÞ þ qðxÞy ¼ 0 ð8Þ 2 dx dx with various levels of approximation. In this equation, the variable coefficients p(x) and q(x) are most likely polynomials represented by power series. An approximate solution of Eq. (8) is viable in terms of a power series of the following form yðxÞ ¼
1 X i¼0
ai ðx x0 Þi ¼ a0 þ a1 ðx x0 Þ þ a2 ðx x0 Þ2 þ
ð9Þ
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where a0, a1, a2, . . . are the coefficients and x0 is the center of the power series. The theory specialized on the existence, uniqueness, and convergence of power series solutions may be found in Ince [7]. Since p(x) = 0 and q(x) = M2R is a monomial in Eq. (1), the power series method is appealing for solving this particular equation. Accordingly, the following summation: 1 X 2 m i ai ðR 1Þ ð10Þ hðRÞ ¼ a0 þ a1 ðR 1Þ þ a2 ðR 1Þ þ þ am ðR 1Þ ¼ i¼0
seems to be adequate to accomplish this task. For convenience, the fin tip, R = 1, is taken as the center of the power series. As a result, Eq. (10) reveals that the first coefficient a0 delivers the dimensionless tip temperature ht = h(1) = a0 right away. After calculating the second derivative of h(R) from Eq. (10), substituting the expression in Eq. (1) and carrying out the algebraic manipulations, we get the following system of two algebraic equations for the collection of coefficients, ai: i¼0:
2a2 M 2 a0 ¼ 0
i ¼ 1; . . . 1 :
ði þ 2Þði þ 1Þaiþ2 M 2 ðai þ ai1 Þ ¼ 0
ð11aÞ ð11bÞ
First, the boundary condition of negligible heat loss at the fin tip given by Eq. (2b) nullifies the second coefficient a1 of Eq. (10): a1 ¼ 0
ð12Þ
Second, the boundary condition of prescribed temperature at the fin base in Eq. (2a) brings forward an additional algebraic equation for the remaining coefficients, ai (i51): / X i ai ð1 cÞ ¼ 1 ð13Þ i¼0
Conceptually, the exact calculation of the general term in Eq. (10) would lead us to the exact solution having combinations of modified Bessel functions as seen in Eq. (6). However, we are trying to avoid this tortuous route in this work. To fulfill our definitive goal, we begin with the truncation of the power series by taking a finite number of terms from the sequence of Eqs. (10)–(13). In this methodical way, approximate expressions for the coefficients may be quickly deduced: (1) the coefficients ai are represented in terms of the first coefficient a0 (the dimensionless tip temperature) and M2 (the enlarged Biot number) and (2) the first coefficient a0 (the dimensionless tip temperature) depends on M2 (the enlarged Biot number) and the normalized radii ratio c. Unfortunately, under the premises of this generic format it is not possible to assess the convergence of the truncated power series. In fact, the convergence patterns of the truncated power series may vary from case to case depending on the values assigned to c and M2. For the examination of individual case studies, the values of M2 and c are determined from the specification of the geometrical, hydrodynamic and thermal quantities. In this sense, it may be demonstrated later that
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Table 1 General expressions from the second to the tenth coefficient of the power series a2 ¼
M2 a0 2!
M2 a0 3! 2 M þ1 a6 ¼ M 4 a0 6! 2 4M þ 7 a9 ¼ 4M 6 a0 9! a3 ¼
4M 4 a0 5! 2 M þ 28 a8 ¼ M 6 a0 8! a5 ¼
a4 ¼
M4 a0 4!
9M 6 a0 7! 2 M þ 100 ¼ M 8 a0 10!
a7 ¼ a10
the number of terms that must be retained to guarantee adequate levels of accuracy for h(R) grows with higher values of M2 and smaller values of c. After applying first Eq. (11) for i = 0, . . . , 8, the ensuing expressions for the coefficients a2, . . . , a10 expressed in terms of a0 and M2 are itemized in Table 1. Second, the leading coefficient a0 may be then calculated with Eq. (13). Next, in order to compute the fin efficiencies within the platform of the power series method, we substituted the approximate temperature distributions, h(R), as given by Eqs. (10)–(13) into the pair of Eqs. (5a) and (5b). Specifically, for a power series with m terms, the respective expressions for g are given by gdif
P 2 mi¼2 i ai ðc 1Þi2 ¼ M 2 ð1 þ cÞ
ð14Þ
and 2 gint ¼
Pm
i i¼0 ai ðc 1Þ
ð1 þ cÞ
c1 iþ2
1 þ iþ1
ð15Þ
Here, we called gdif the fin efficiency calculated by differentiation and gint the fin efficiency calculated by integration. As may be confirmed later, the above approximate g-relations are evaluated easily with the substitution of a finite number of coefficients ai presented in Table 1, in conjunction with the leading coefficient a0 as given by Eq. (13). It should be pointed out that by virtue of the two calculation options available for g, the integration approach via Eq. (15) yields better results than the differentiation approach via Eq. (14). The truncated power series of Eq. (10) is now complete and may be used with confidence for the computation of the temperature distributions (including tip temperatures) and fin efficiencies of any annular fin of hyperbolic profile. As an illustrative example, we report in the next section two sample calculations that convey the mechanics of the power series method retaining 5 and 10 terms for a pre-selected combination of c and M2. Incidentally, in the event that more than 10 coefficients ai need to be employed in the power series, any symbolic computer code or even a spreadsheet may facilitate the required
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numerical calculations, since it is only a question of solving a system of linear algebraic equations.
5. Practical example Observation of the fin efficiency diagram in Ref. [4] reveals that the modified Biot number M2 in the abscissa goes from 0 to 10. In addition, the normalized radii ratio c being a parameter extends from a small value of c = 1/4 (long annular fin of hyperbolic profile) to the limiting large value of c = 1 (straight fin of rectangular profile). The format of the fin efficiency diagram insinuates that the largest outer-to-inner radii ratio normally used for annular finned tubes, stays around 4. In view of this variability, the choice of c = 1/4 identifying the bottom curve in the fin efficiency diagram in [4] seems to be a logical choice because it represents the worst possible scenario in real applications of heat exchange devices [1,2]. For the specific fin size dictated by c = 1/4, we selected an intermediate value of the modified Biot number, say M2 = 6.572, so that the annular fin of hyperbolic profile operates with a fin efficiency g of a moderate magnitude 0.520. Now, the implementation of the power series method is undertaken systematically with the goal at calculating the temperature distribution along the fin and the heat transfer rate from the fin to the fluid. Let us begin with a 5-term power series as delineated in the preceding section. Doing the algebra, the approximate dimensionless temperature distribution h5(R) results in h5 ðRÞ ¼ 0:38255 þ 1:25710ðR 1Þ2 þ 0:41902ðR 1Þ3 þ 0:68845ðR 1Þ4 þ 0:55076ðR 1Þ5 ð16Þ Doubling the number of terms in a second try, the approximate dimensionless temperature distribution h10(R) based on a 10-term power series turns out to be h10 ðRÞ ¼ 0:37329 þ 1:2266ðR 1Þ2 þ 0:40888ðR 1Þ3 þ 0:67178ðR 1Þ4 þ 0:53743ðR 1Þ5 þ þ0:23674ðR 1Þ6 þ 0:18921ðR 1Þ7 þ 0:09085ðR 1Þ8 þ 0:03888ðR 1Þ9 þ 0:02045ðR 1Þ10
ð17Þ
Fig. 2 displays the closeness of the two approximate temperature distributions h5(R) and h10(R), as compared against the exact temperature distribution hexact(R) coming from Eq. (6). Herewith, it may be confirmed that both the 5- and the 10-term series handle the temperature variation along the annular fin of hyperbolic profile admirably. The largest error in the predicted temperature occurs at the fin tip, as could have been expected. The resulting two approximate tip temperatures are ht = h5(1) = 0.38255 and ht = h10(1) = 0.37329. Using as a reference the exact tip temperature hexact(1) in Eq. (6), h10(1) overlaps with hexact(1) whereas h5(1) exhibits an error of order 2.5% with respect to hexact(1). To guarantee accuracy, the exact temperatures and the exact fin efficiencies in Eqs. (6) and (7) were evaluated numerically with the symbolic code Maple [8].
I. Arauzo et al. / Applied Thermal Engineering 25 (2005) 623–634
631
θ 1 θ θ _5 term θ _10 term
0.9
0.8
0.7
0.6
0.5
0.4
0.3 0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
R
Fig. 2. Approximate temperature distributions for M2 = 6.572 and c = 1/4, using a 5- and a 10-term power series.
6. General cases To assess the convergence of the 10-term power series temperature distribution, h10(R), the crucial test involves the approximate tip temperatures. Hence, h10(1) in Eq. (17) along with the exact tip temperatures hexact(1) in Eq. (6) are compared for all possible combinations of the fin parameter n and the normalized radii ratio c. The outcome of the comparison is displayed in Fig. 3. It is observable in the figure that the agreement is excellent, the only subtle deviations take place in the limiting condition related to large n and small c, i.e., n = 3 and c = 0.1. Further, it should be recognized that the uppermost curve for c = 1 is actually the limiting situation of the annular fin of hyperbolic profile symbolizing a straight fin of rectangular profile (r2 r1)/r2 ! 0. Herewith, the tip temperature is calculated by the ratio: hð1Þ ¼
1 pffiffiffi coshð 2nÞ
ð18Þ
The exact fin efficiency curves are obtained from the evaluation of Eq. (7) for all combinations of c and M2. The corresponding numbers are compared against the approximate fin efficiencies computed by means of Eq. (15) united to the 10-term temperature distribution h10(R) in Eq. (17). It should be mentioned that the first curve for c = 1 is actually the limiting case of a straight fin of uniform profile, (r2 r1)/r2 ! 0 whose fin efficiency is calculated as: pffiffiffi tanhð 2nÞ pffiffiffi g¼ ð19Þ 2n
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I. Arauzo et al. / Applied Thermal Engineering 25 (2005) 623–634 θ(1)
1 0.9 0.8 0.7 0.6 0.5 0.4 c=0.5 c=0.7 c=1
0.3 0.2
c=0.1
c=0.2 c=0.3
0.1 0 0
0.5
1
1.5
2
2.5
3
ξ
Fig. 3. Approximate tip temperatures using a 10-term power series (black dots) and the exact tip temperatures (solid lines).
The agreement between the exact and the approximate estimates of g is perfect for all radii ratios contained in the large interval 0.2 6 c 6 1 and values of the fin parameter n that lie in the interval 0 6 n 6 2. These combinations of c and n cover an ample gamma of fin efficiencies that extend between 0.25 and 1. For the fin holding a radii ratio c = 0.1 the last curve indicates that the agreement is also perfect but it is applicable up to n = 1.5 inclusive. To save journal space, we decided not to include the differential-based calculations of g utilizing a 10-term power series, i.e. Eqs. (14) and (17). From the framework of thermal performance per se, it is known that regardless of the cross sectional shape, the addition of fins is beneficial when they deliver efficiencies with magnitudes near one [1,2]. In other words, this statement is indicative of efficiency numbers that lie in the upper left region of Fig. 4. For completeness, it should be added that the 5-term series in Eq. (16) gives reasonably good results for the same values of c coupled with values of n up to 1 and even up to 1.5 for c P 0.5. Contrary to the trends manifested in Fig. 4, the fin efficiencies based on 5-term series tend to be slightly overestimated. Owing to the detailed characterization of dimensionless radii ratios c, it may be categorically stated that the complete fin efficiency diagram of Fig. 4 containing six curves supersedes the reduced fin efficiency diagram of Ref. [4] that contains only three curves.
I. Arauzo et al. / Applied Thermal Engineering 25 (2005) 623–634 η
1 0.9 0.8 0.7
c=0.1
η_int c=0.1
c=0.2
η_int c=0.2
c=0.3
η_int c=0.3
c=0.5
η_int c=0.5
c=0.7
η_int c=0.7
633
c=1
0.6 0.5 0.4 0.3
c = 1.0 c = 0.7 c = 0.5 c = 0.3 c = 0.2 c = 0.1
0.2 0.1 0 0
0.5
1
1.5
2
2.5
3
ξ
Fig. 4. Approximate fin efficiencies using a 10-term power series (black dots) and the exact fin efficiencies (solid lines).
7. Conclusions The major conclusion that may drawn from this technical paper is that a 10-term power series is splendid for predicting flawlessly the temperature distributions and the companion heat transfer rates of annular fins of hyperbolic profile of practical significance in heat transfer engineering. Compared to the exact temperature distribution and the exact heat transfer rates evaluated with intricate Bessel functions of fractional order, it was realized that the 10-term power series solution is versatile, offers considerable economy and more importantly supplies accurate results. The analytic expressions for the participating series coefficients are compact, permitting the study of a wide range of realistic combinations of the two controlling parameters: the normalized radii ratio c and the enlarged Biot number M2. In principle, the power series method being the precursor of the venerable Bessel functions may be extended with ease to a general class of straight or annular fins of tapered profile in which the common straight or annular fin of rectangular profile are just particular cases. Acknowledgments The authors are indebted to Richard D. Kirkham, College of Engineering, Idaho State University, Pocatello, Idaho for his help during the early stages of the calculations with the power series method.
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References [1] A.D. Kraus, A. Aziz, J.R. Welty, Extended surface heat transfer, Wiley, New York, 2000. [2] R.L. Webb, Principles of enhanced heat transfer, Wiley, New York, 1994. [3] E. Schmidt, Die wa¨rmeu¨bertragung durch rippen, Zeitschrift des Vereines Deutscher Ingenieure 70 (1926) 885–889, pp. 947–951. [4] P.J. Schneider, Conduction heat transfer, Addison-Wesley, Reading, MA, 1955. [5] V. Arpaci, Conduction heat transfer, Addison-Wesley, Reading, MA, 1966. [6] W.E. Boyce, R.C. DiPrima, Elementary differential equations and boundary value problems, Wiley, New York, 1965. [7] E.L. Ince, Ordinary differential equations, Dover, New York, 1956. [8] E. Kamerich, A guide to maple, Springer-Verlag, New York, 1998.