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Exact and Approximate Analytic Methods to Calculate Maximum Heat Flow in Annular Fin Arrays with a Rectangular Step Profile

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Abstract

Exact and approximate techniques to determine the performance of annular fins with a step rectangular profile are developed. A simple approximate method is proposed to analyze the prevalent heat transfer characteristics of the fin array. An algebraic expression based on the mean value approach is used as an approximation tool, and the temperature distribution in the fins is determined using an exponential function. A method based on a modified Bessel function formulation is employed for exact analysis. The analyses are extended to optimize fins based on the principle of maximizing heat transfer rate for a given volume. The results obtained from the exact and approximate analyses are presented in a one-to-one comparative manner to allow for a wide range of practical design variables. The error in the approximate analysis calculations is investigated, and it is found to be well within engineering accuracy requirements. It is expected that the approximate analytical tool designed here will be extremely useful for designers who want to easily determine design parameters. Because the approximate methodology is so simple, all calculations can easily be done.

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Abbreviations

Bi :

Biot number based on the fin-side convective heat transfer coefficient, h r 1/k f

Bi 1 :

Biot number based on the tube convective heat transfer coefficient, h L r 0/k w

D :

Differential operator, see Eq. 14

D −1,D 0, . . . ,D 4 :

Dimensionless parameters in Eq. 17

h :

Convective heat transfer coefficient on the fin side areas (W. m−2. K−1)

h L :

Convective heat transfer coefficient on the inner side tube (W.m−2.K−1)

I m (Z):

Modified Bessel function of first kind of order m and argument Z

k f :

Thermal conductivity of the fin material (W. m−1.K−1)

k w :

Thermal conductivity of the tube material (W. m−1. K−1)

K m (Z):

Modified Bessel function of second kind of order m and argument Z

q :

Actual heat transfer rate through the symmetric sector of a fin-array (W)

Q :

Dimensionless actual heat transfer rate, q/4π k w r 0 (T L −T a )

q f :

Actual heat transfer rate through a fin (W)

q fi :

Ideal heat transfer rate through a fin (W)

Q f :

Dimensionless actual heat transfer rate, q f/4π k w r 0 (T L −T a )

Q fi :

Dimensionless ideal heat transfer rate, q fi/4π k w r 0 (T L −T a )

q w :

Heat transfer rate through the symmetric module of base area with no fin condition (W)

Q w :

Dimensionless heat transfer rate, q w/4π k w r 0 (T L −T a )

r :

Radial coordinate starting from the center of the tube (m)

\({\bar{r}}\) :

Mean radius (m)

r 0 :

Inner radius of the tube (m)

r 1 :

Outer radius of the tube (m)

r 2 :

Step radius (m)

r 3 :

Outer radius of a step fin-array (m)

R :

Dimensionless radial coordinate, r/r 3

R 1 :

Dimensionless outer radius, r 1/r 3

R 2 :

Non-dimensional step radius, r 2/r 3

\({\bar{r}}\) :

Dimensionless mean radius, see Eq. 15, \({{\bar{r}}/{r_3}}\)

T f1:

Local fin surface temperature for larger thickness (K)

T f2:

Local fin surface temperature for smaller thickness (K)

T a :

Ambient temperature (K)

T fb :

Fin temperature at the base (K)

T L :

Fluid temperature inside the tube (K)

T w :

Local wall temperature (K)

U :

Dimensionless fin volume, V/2π r 30

V :

Fin volume (m 3)

Z 1, Z 2 :

Dimensionless fin parameters defined in Eq. 3

α :

Thermal conductivity ratio, k f/k w

β :

Dimensionless pitch length, P/r 1

δ :

Thickness ratio, δ 2/δ 1

δ 1 :

Semi-base thickness (m)

δ 2 :

Semi-tip thickness (m)

\({\varepsilon _{\rm f}}\) :

Fin effectiveness

γ :

Heat transfer coefficient ratio, h/h L

ω :

Dimensionless wall temperature, (T w −T a )/(T L −T a)

\({\phi }\) :

Dimensionless fin temperature, (T f2 −T a ) /(T L −T a )

\({\Phi}\) :

Augmentation factor

η f :

Fin efficiency

\({\psi }\) :

Fin aspect ratio, δ 1/r 1

θ :

Dimensionless fin temperature, (T f 1 −T a )/(T L −T a )

θ b :

Dimensionless fin base temperature, (T fb −T a )/(T L −T a )

τ 1, τ2, . . . ,8 :

Dimensionless variables defined in Eqs.6 and 19

Λ :

Notation defined in Eqs. b and 19b

Opt:

Optimum

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Authors and Affiliations

  1. School of Mechanical Engineering, Hanyang University, 17 Haengdang-dong, Sungdong-gu, Seoul, 133-791, Republic of Korea

    Balaram Kundu & Kwan-Soo Lee

  2. Department of Mechanical Engineering, Jadavpur University, Kolkata, 700032, India

    Balaram Kundu

  3. Department of Mechanical Engineering, The University of Texas at San Antonio, San Antonio, TX, 78249, USA

    Antonio Campo

Authors
  1. Balaram Kundu
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  2. Kwan-Soo Lee
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  3. Antonio Campo
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Correspondence to Kwan-Soo Lee.

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Kundu, B., Lee, KS. & Campo, A. Exact and Approximate Analytic Methods to Calculate Maximum Heat Flow in Annular Fin Arrays with a Rectangular Step Profile. Int J Thermophys 33, 1314–1333 (2012). https://doi.org/10.1007/s10765-012-1271-4

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  • DOI: https://doi.org/10.1007/s10765-012-1271-4

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