Approximate analytic temperature solution for uniform annular fins by adapting the power series method

Int. Comm. HeatMass Transfer, Vol. 25, No. 6, pp. 809-818, 1998

Copyright © 1998 Elsevier Science Ltd Printed in the USA. All rights reserved 0735-1933/98 $19,00 + .00

Pergamon

PII S0735-1933(98)00067-0

A P P R O X I M A T E A N A L Y T I C T E M P E R A T U R E S O L U T I O N FOR U N I F O R M A N N U L A R FINS BY A D A P T I N G T H E P O W E R SERIES M E T H O D

Antonio Campo, Franklin Rodrfguez College of Engineering Idaho State University, Pocatello, ID 83209

( C o m m u n i c a t e d b y J.P. H a r t n e t t a n d W.J. M i n k o w y c z )

ABSTRACT The primary specification faced by thermal engineers when designing annular fins is the amount of heat transfer from a finned tube to a fluid. The temperature along annular fins of uniform thickness is governed by an ordinary differential equation of second order with variable coefficients, called the modified Bessel equation of zero order. Approximate temperature distributions and fin efficiencies, both of good quality, have been obtained by adapting a power series method for solving this Bessel equation with symbolic algebra software such as Maole or Mathematica. Students of heat transfer courses can benefit from this simple computational procedure that circumvents the use of 8essel functions and still produces approximate analytic results of good caliber. © 1998ElsevierScienceLtd

Introduction The temperature variation along uniform straight fins is governed by an ordinary differential equation of second order with constant coefficients. Due to the nature of the constant coefficients, exact analytic prediction of the thermal characteristics (temperatures and heat transfer rates) of straight fins is relatively easy because it requires numerical evaluations of standard hyperbolic functions. course, these operations are uncomplicated and can be done with a calculator.

Of

Conversely, the

temperature variation along uniform annular fins is governed by an ordinary differential equation of second order having variable coefficients. The dimensionless version of this equation is called the modified Bessel equation of zero order and clearly its exact analytic solution is involved due to the presence of variable coefficients.

Harper and Brown [1] utilized this equation in connection to the

thermal analysis of aiPcooled engines.

Focusing on aspects of thermal design of fins, Gardner [2] devised a compact format for presenting the dimensionless heat transfer by plotting the fin efficiency, r/, in the ordinate and the dimensionless thermo-geometric parameter, y2, in the abscissa. The resulting family of curves was 8O9

810

A. Campo and F. Rodriguez

Vol. 25, No. 6

parameterized by the radii ratio c defined as rJrl, where the upper bounding curve represented a uniform straight fin (c = 1 ). This graphical representation was a definite breakthrough at that time and led to the construction of the popular fin efficiency diagram (see Cengel [3]).

This technical note addresses an elementary computational technique for an approximate analytic solution of the heat conduction equation in an annular fin, the modified Bessel equation of zero order, resorting to a combination of a curve fit procedure and the standard power series method. As is demonstrated later, this coupling facilitates the rapid determination of temperature variations, as well as fin efficiencies of uniform annular fins in terms of the t w o controlling parameters, c and y2. Beyond any doubt, the primary objective here is to bypass the Bessel functions and its operations, a result which will please students in undergraduate courses on heat transfer.

Mathematical Model Steady-state heat transfer in a uniform annular fin of thickness 2t, inner radius r 1, outer radius r~, and constant properties is modeled by the heat conduction equation in cylindrical coordinates:

d20

+

dR 2

1

dO

R

dR

y2 8 = 0

(1)

This dimensionless equation is classified as an homogeneous ordinary differentialequation of second order with variable coefficients, called the modified Bessel equation of order zero (see Kreyszig [4]). Note that the variable coefficient is given by 1/R.

The boundary conditions associated with a prescribed temperature at the base, and zero heat loss at the tip are (9 = 1

at

R

= c

and

dO dR

_ 0

at

R = 1

(2)

In the above equations, the dimensionless variables are 8 and R, whereas c is the radii ratio and y2 is the dimensionless thermo-geometric parameter.

Once the temperature distribution, e(R), has been obtained by any solution method, the heat transfer from any fin to a fluid can be computed in t w o ways: differentiating 8(R) at the fin base or integrating 8(R) along the fin. Following [2], calculation of the heat transfer from a uniform annular fin, O, has been traditionally done by w a y of the dimensionless heat transfer or fin efficiency, r/ = Q/Q=, where Q~ designates the ideal heat transfer rate from a fin possessing identical characteristics, but infinite thermal conductivity. Hence, the corresponding relations via differentiation and integration are

Vol. 25, No. 6

TEMPERATURE SOLUTION FOR UNIFORM ANNULAR FINS

-2c~

~-~

I"I -

and

y~

respectively.

811

1"I -

(1 - c 2)

(i - c 2)

Logically, employing an exact 8(R) the t w o expressions in eq. [3) supply the

same

numerical result, since the heat loss from the fin tip is zero. In contrast, it may be anticipated that utilization of an approximate 8(R) may exhibit some discrepancies in the t w o numerical results.

Additionally, the tip temperature, 8 t = 8(1), cannot be overlooked for safety reasons.

This

temperature has to be within certain bounds to avoid burns of technical personnel working in plant environments (see Muir et al. [5]). When the tip temperature is at issue, it must be remembered that the higher the fin efficiency the higher the tip temperature.

Exact Method

The exact analytic solution of eqs. [1) and (2) can be taken directly from Mills [6]. That is

0 ( a ) = KI(y) Z0(ya) + A ( 7 ) K0(Ym Kz (Y) ~0 (yc) * A (Y) K0 (Vc)

(4)

where I V (*) and K v (*) are defined in the Nomenclature,

Adaotation of the Power Series Method

The p o w e r series method is widely k n o w n as a robust, standard method for solving linear ordinary differential equations with variable coefficients of the general type [4]: y"

+ p(x)y

I + q(x)y

= 0

(5)

In this equation, the functions p[x) and q(x) are most likely polynomials which are represented by p o w e r series in powers of x - Xo. Thereby, a solution of eq. (5) may be assumed in terms of p o w e r series of the form

y(x)

= ~,a=(X-Xo)"

= a o + al (x-x

o)

+ a= ( X - X o )

2 + ...

(6)

ar.=O

where the constants a., a,, a, .... are the coefficients of the series, x is the independent variable, and Xo is the center of the series. The theory specialized on the existence, uniqueness, and of the p o w e r series method can be found in Ince [7].

convergence

812

A. Campo and F. Rodriguez

Vol. 25, No. 6

Inspection of the family of curves in the fin efficiency diagram of [3] reveals that the radii ratio, c = r,/r 2, takes finite values of 1 (straight fin), 0.5, 0.33, 0.25, and 0.2. On the other hand, it is also known from geometrical considerations that the base of an uniform annular fin coincides with the outer surface of a circular tube, meaning that the dimensionless radius of the fin R, never reaches zero. Consequently, eq. (1) does not possess a regular singular point at the tube centerline, R = 0. Since the smallest value of c is 0.2, the inverse of the dimensionless radial variable, 1/R, responsible for the variability of the coefficient in the second term of eq. (1), can be conveniently replaced by an equivalent polynomial, p(R), inside the pertinent interval for c: [0.2,1], a closed interval.

In principle, this

observation is advantageous because it enables us to seek an approximate analytic solution of eq. (1) by means of the standard power series method [4], instead of resorting to an exact Bessel function method; the latter is a generalization of the Frobenius series method [4].

Among the polynomial fits capable of handling the numerically-evaluated function l/R, the polynomial of order 4: !

R

~ p

(R)

=

11.711389

- 49.782752R

+ 97.112966R

2

(7) -

88.020267R

3 +

29.991259R

4

possesses a very high correlation coefficient, R2 = 0.9995. The polynomials of order 2 and 3 produced R' = 0.9687 and 0.9956, respectively.

Accordingly, the combination of eqs. (1) and (7) results in the transformed differential equation

d28 + p(R) d R ~.

dO

_

y2

0

=

0

(8)

--~

and the boundary conditions given by eq. (2) are unaffected. Now, comparing eqs. (5) and (8) on a one-to-one basis discloses that p(x) = p(R) is the sought equivalent polynomial, whereas q(x) reduces to a constant - ),2, the minimal expression of a polynomial.

APProximate Analytic Results

A collection of approximate temperature distributions of uniform annular fins have been successfully determined by blending a fitted polynomial, a power series method and a symbolic algebra software such as Maple or Mathematica on a desktop computer. A group of representative results has been generated for pairs of the t w o intervening parameters: the radii ratio c and the dimensionless thermo-geometric parameter, ),2 Due to space limitations only the temperature variations connected to the most critical cases described by combinations of c = 0.2 and ),2 = 0.5, 1.5, 3 and 10 have been reported here.

Certainly, the choice of the smallest c = 0.2 is deliberate because it identifies the

Vol. 25, No. 6

TEMPERATURE SOLUTION FOR UNIFORM ANNULAR FINS

813

bottom curve in the fin efficiency diagram [3], which is associated with a long annular fin (r= = 5 r 1) and four different values of the abscissa, y = r2(hr2/kt)l~L Implementing p o w e r series of order 2 and higher yields approximate analytic temperature distributions of the form 8(R)

= a o + a z (R-Z)

+

...

+ a s (R-I)

s

(9)

where the center of the series, Ro -- 1 (the fin tip), is chosen by the symbolic algebra code.

For

instance, for the union of c = 0.2 and 1,2 = 10 (the w o r s t case in the aforesaid list), the temperature distribution leads to 0(R)

= 0.I05370

+ 0.526850

+ 0.453553

(R-Z)

(R-Z)

2 - 0.177829

~ - 0.856391

(R-I)

(R-I)

3

(1o)

s

A "ratio test" guarantees that the interval of convergence of this p o w e r series is 0.2 s R ~ 1. Also,

it should be mentioned that this type of Taylor series expansion is beneficial for evaluating the tip temperature, 0t = 8(1 ), because its magnitude is delivered by the independent term of the p o w e r series since the remaining terms vanish. Hence, the tip temperature in eq. (10) is 8t = 0 . 1 0 5 3 7 0 .

Further, for purposes of comparison, the exact temperature distribution, eq. (4), is evaluated for a poor fin with c = 0.2 and 1,2 = 10: e(a)

= 0.009710

z0(VYdm

+ 1.342805

Ko(V~R)

(11)

This relation constitutes the baseline solution owning an efficiency in the neighborhood of 20 %.

TABLE 1 Comparison of CPU Times (in seconds)

y2

Power series of order 5

Bessel functions

0.5

1.582

3.685

1.5

1.652

3.765

3

1.693

3.936

10

1.712

4.306

In general, the exact temperature distribution, eq. (4), once particularized for pairs of c = 0.2 and variable i~ provides the baseline solution. The corresponding exact and approximate temperature

814

A. Campo and F, Rodriguez

Vol. 25, No, 6

0`116

0 0`9-

0.~

02

R

I-

0 0.8~

0.7"

0.1~

0.5

, 0.2

,

=

1.

,

~

appr I 0,4

~

,

,

,

I 0`6

~

~

'

,

I

J

~

*

OJS

R FIG. 1 Comparison between approximate and exact temperature distributions for c = 0.2 and four different values of y=.

(Figure continued.)

Vol. 25, No. 6

TEMPERATURE SOLUTION FOR UNIFORM ANNULAR FINS

o.e

exact

approx

,

i

J

i

J

J 0.6

O.4

0.2

O~

t

R

0.8

rox 0.6

exact 0.4

0.2

J 0.2

,

i

i

I n4

i

i

i

p

I O.e

i

,

i

i

R FIG. 1 (Continued)

J

I 0.8

i

,

815

816

A. Campo and F. Rodriguez

distributions are omitted for brevity.

Vol. 25, No. 6

The CPU times related to the symbolic calculations of each

temperature distribution are listed in Table 1. The CPU times for the exact Bessel function solution, eq. (4), are consistently 2.5 times greater than the CPU times dealing with the approximate power series of order 5, eq. (10).

The goodness of the approximate temperature distributions, eq. (10)0 can be judged in Fig. 1, and their coincidence with the exact temperature distributions, eq. (4), is striking. Overall, the usual temperature deviation is around 0.01 units and the maximum temperature deviation does not exceed 0.02 units.

Needless to say, the accuracy of the power series temperature distribution may be

enhanced by adding more terms to the series.

For heat transfer calculations, it should be remembered that the equality of fin efficiencies by differentiating and integrating eq. (3) holds true for exact temperature distributions only. Inasmuch as the power series method supplies an approximate solution, it has been recommended by Arpaci [8] that the fin efficiencies computed by integration are consistently more accurate than the ones computed by differentiation.

Nevertheless, results by both approaches have been reported in the following

paragraphs. Numerical values of the fin efficiencies are presented in Table 2 for c = 0.2 (fixed) and variable y2 = 0.5, 1.5, 3, and 10. These combinations supply numerical values for the fin efficiency, q, in the vicinity of 80 %, 60 %, 40 % and 20 %, respectively. Beyond doubt, the fin efficiencies obtained by the integral route are consistently more accurate than those determined by way of the derivative. Specifically, the integration approach connected to a poor fin whose efficiency is around 20 % (that is for c = 0.2 and y2 = 10} produced a maximum error of 11.7 %. For more realistic cases involving better fins with higher efficiencies (say r/ > 80 %) the error by integration was always below a reasonable margin of 1.7 %.

It is important to add that the standard fin efficiency diagram [3] portrays curves for cases characterized by combinations of radii ratios contained in the closed interval 0.2 ~; c < 1 with values of the dimensionless thermo-geometric parameter, y=. Surprisingly, the diagram does not include the curves connected to situations described by c < 0.2 and very low ~ .

These cases, located in the

upper left corner of the diagram, supply qualitative efficiencies in excess of 80 %.

Consequently,

extrapolation in this region from the given curves is questionable and may lead to erroneous results. Definitely, the actual numbers for the ensuing high efficiencies may be easily determined with the approximate solution method developed in this paper enabling a precise estimation of heat transfer rates.

The optimum dimensions of annular fins of uniform thickness are those dimensions that give the greatest amount of heat transfer, Q~,.~, for a given quantity of material (fixed t, rl, and r2). This

Vol. 25, No. 6

TEMPERATURE SOLUTION FOR UNIFORM ANNULAR FINS

817

information, presented for the first time by Jakob [9] in 1949 in the form of a nomogram, is invariant with the solution method employed.

TABLE 2 Comparison of the Fin Efficiencies

Procedure

y=

Power series of order 5 (error)

Bessel functions (exact)

derivative

0.5

0.8007 (1.5%)

0.8127

integral

0.5

0.7988 (1.7%)

0.8127

derivative

1.5

0.5563 (7.1%)

0.5985

integral

1.5

0.5842 (2.4%)

0.5985

derivative

3

0.3839 (12.2%)

0.4372

integral

3

0.4332 (0.9%)

0.4372

derivative

10

0.1611 (24.7%)

0.2140

integral

10

0.2391 11.7(%)

0.2140

In conclusion, prediction of the temperature changes and companion heat transfer rates of uniform annular fins can be obtained in an accurate fashion by linking a curve fit (replacing the variable coefficient 1/R in eq. (1)) with a power series method of order 5, at least. It may be inferred that thermal results of comparable quality must be obtained for higher radii ratios (c > 0.2) regardless of the value of I~. Unquestionably, the proposed combined technique exceeds expectations, and due to its marked simplicity may be adopted for instruction of the important topic of annular fins in basic courses on heat transfer.

Nomenclature

c

radii ratio, r;/r 2

h

average heat transfer coefficient

I~(*)

modified Bessel function of first kind and order v

818

A. Campo and F. Rodriguez

Kv(*)

modified Bessel function of second kind and order v

k

thermal conductivity

(3

heat transfer rate

Qi

ideal heat transfer rate

r

radial coordinate

rl

inner radius

r2

outer radius

R

dimensionless r, r/r 2

t

semi-thickness

T

temperature

Tb

base temperature

Tt

tip temperature

T=

fluid temperature

B

thermo-geometric parameter, (h/kt) "~

Y

dimensionless #, #r 2

t7

fin efficiency or dimensionless Q, Q/Q~

8

dimensionless T, (T - T®)/(T b - T®)

Vol. 25, No. 6

Acknowledgements The authors appreciate the suggestion of Prof. Theodore F. Smith (The University of Iowa) with regards to the inclusion of the CPU times for the exact and approximate solution methods.

References 1.

Harper, W.P. and Brown, D.R., Mathematical equations for heat conduction in the fins of aircooled engines, NACA Report 158 (1922).

2,

Gardner, K.A., Efficiency of extended surfaces, Trans. ASME67, 621 (1945).

3.

Holman, J.P., Heat Transfer, 8th. ed., McGraw-Hill, New York, NY (1997).

4.

Kreyszig, E., Advanced Engineering Mathematics, 7th. ed., Wiley, New York, NY (1993),

5.

Muir, I.F.K., Barclay, T.L. and Settle, J.A.D., Burns and their Treatment, Butterworths, Boston, MA (1987).

6.

Mills, A.F., Heat Transfer, Irwin, Boston, MA (1992).

7.

Ince, E.L., Ordinary Differentia/Equations, Dover, New York, NY (1956).

8.

Arpaci, A., Conduction Heat Transfer, Addison-Wesley, Reading, MA (1966).

9.

Jakob, M., Heat Transfer, Vol. I, Wiley, New York, NY (1949).

Received April 28, 1998