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Transient Response of a Spiral Fin with its Base Subjected to the Variation of Heat Flux



J.S. Wang, W.J. Luo and S.P. Hsu
 
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ABSTRACT

The problem of transient response of a spiral fin, with its end insulated, is analyzed with the base end subjected to a variation of fluid temperature. The method of Laplace transforms and with a technique derived by the Keller and Keller, of which has an exponential like form, is applied to study the transient response. Both of a unit step change and a sinusoidal temperature change are analyzed. Moreover, the results of the temperature distribution and the heat flux at the base of the fin are obtained. Solutions are developed for both small and large values of time. Typical results are presented in both tables and graphs. Comparisons are made and the results show a good agreement in the physical circumstance.

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  How to cite this article:

J.S. Wang, W.J. Luo and S.P. Hsu, 2008. Transient Response of a Spiral Fin with its Base Subjected to the Variation of Heat Flux. Journal of Applied Sciences, 8: 1798-1811.

DOI: 10.3923/jas.2008.1798.1811

URL: https://scialert.net/abstract/?doi=jas.2008.1798.1811
 

INTRODUCTION

The problem of response in fins has been of much interest for many researchers and engineers due to its magnificently industrial applications. The use of Fins to enhance the heat dissipation from a hot surface is very extensive in many areas of engineering applications. Besides the traditional applications, such as power generator, plants and vehicles, fines are also used in heat removal devices foe electronic components. Park et al. (2007) used the pin-fin type heat sinks for different fin shapes to enhance the heat transfer of a heat sink and the optimum values of the design variables such as fin height, fin width or fin diameter and fan-to-heat sink distance at the junction of a heat sink and a heat source are investigated. T'joen et al. (2007) applied an experimental study to investigate the performance of a fin-and-tube heat exchanger in two different configurations.

In a conventional heat exchanger heat is transferred from one fluid to another through a metallic wall. The rate of heat transfer is directly proportional to the extent of the wall surface, the heat transfer coefficient and to the temperature difference between one fluid and the adjacent surface. It might be expected that the rate of heat transfer per unit of the base surface area would increase in direct proportion. However, the average surface temperature of the fins tends to decrease approaching the temperature of the surrounding fluid so the effective temperature difference is decreased and the net increase of heat transfer would not be in direct proportion to the increase of the surface area and may be considerably less than that would be anticipated on the basis of the increase of surface area alone. The performance of fin under steady state conditions has been studied in considerable detail but the transient response of such surfaces to changes in either base temperature or base heat flux has not received much attention. Both of one-dimensional and two-dimensional circular fin have been studied broadly. Chu et al. (1982, 1983a, b) has applied the Fourier series inversion technique to determine the transient response of two-dimensional straight fins and circular fins, one-dimensional annular fin and the composite straight fins. His results showed a good agreement in the physical circumstances. The transient temperature response of the annular fins, a special case (pitch equal zero) for the spiral fin, was well studied. Cheng et al. (1994, 1998) studied the transient response of annular fins of various shapes subjected to constant base heat fluxes. In their work, the inverse method was applied. Yu and Chao-Kuang Chen (1999) applied the Taylor transformation to the transient temperature response of annular fin. When the end of the fin is not insulated, Harper and Brown (1992) have shown that, under certain circumstances, an equivalent fin with end insulated can be obtained by suitably increasing its length. It is also assumed that one-dimensional analysis is valid. One-dimensional analysis has been shown to be valid under steady state conditions for small Biot number by Crank and Parker (1996). Performance and optimum dimensions of longitudinal and annual fins and spines with a temperature-dependent heat transfer coefficient have been presented by Laor and Kalman (1996). In this study, considered the heat transfer coefficient as a power function of temperature and used exponent values in the power function that represent different heat transfer mechanisms such as free convection, fully developed boiling and radiation. The optimum dimensions of circular fins with variable profile and temperature-dependent thermal conductivity have been obtained by Zubair et al. (1996). Campo and Stuffle (1996) presented a simple and compact form correlation that facilitates a rapid determination of fin efficiency and tip temperature in terms of fin controlling parameters for annular fins of constant thickness. Mokheimer (2002) investigated the performance of annular fins of different profiles subject to locally variable heat transfer coefficient. The performance of the fin expressed in terms of fin efficiency as a function of the ambient and fin geometry parameters has been presented in the literature in the form of fin-efficiency curves for different type of fins.

Because fins are frequently used under unsteady conditions, it is worth studying the transient behavior of fins with time-dependent boundary conditions. Otherwise, due to the particular geometry of the spiral fin, the contact surface area of the fin is lager than that of annual fin at the same length. From the through literature survey mentioned above, the effect of temperature-dependent heat transfer coefficient on the fin efficiency of spiral fins has not been investigated. The aim of the present article is to investigate such effects. This type of study would be of direct use by the heat transfer equipment designers and rating engineers. In this study, the solutions of transient response of straight fins are obtained by Laplace transforms that is easy to calculate rapidly convergent approximate solutions for small value of time. After using the Laplace transforms to the differential equation for the temperature of the spiral fin, we can obtain a linear set of second order ordinary differential equation which can be evaluated by employing Keller and Keller’s method (Keller and Keller, 1962; Luo and Yang, 2007; Shih et al., 2008). Solutions of the approximately exponential functions were proposed for the partial differential equation in their papers. This is a novel method in solving the nonlinear parabolic second order differential equation. Therefore, through the combination of Laplace transforms and the method derived by Keller and Keller (1962), we evaluate the transient response of a spiral fin with the base subjected to a variation of fluid temperature.

Fig. 1: The physical model of a spiral fin

Analysis: The physical system consisting of a spiral fin of uniform thickness 2δ, inner radius r1, outer radius r2, pitch P and thermal conductivity k is shown in Fig. 1. The end of the fin, i.e., r = r2, is assumed to be perfectly insulated. It is also assumed that one-dimensional analysis is valid. Initially, the fin is in thermal equilibrium with the surrounding fluid temperature T. At time t = 0, the base temperature is suddenly raised to Tf or subjected to heat flux q0* and from then on, the spiral fin dissipated heat by convection to the environment through a convective heat transfer coefficient h, hf and the properties k, ρ, c of the material of the fin are all assumed to be constant. At the other boundary condition, i.e., at r = r1, the convective boundary condition by ignoring the thermal resistance and capacity of the material in inner wall tube is assumed.

From the conservation of energy, we can get the differential equation of the temperature of spiral fin from the balance of energy, as following:

(1)

where, α = k/ρc is the thermal diffusivity and Biot number is Bi = hfr1/kf.

After introducing the non-dimensional variables,

The dimensionless governing equation is:

(2)

The analysis of the transient response of spiral fin subjected to a variation of fluid temperature will be studied in the two following cases.

Case 1: A unit step change in base fluid temperature: For the case 1, the heat transfer of the fin which base is subjected to a constant temperature is investigated. The ambient temperature is kept constant and no heat sources or sinks are present. The dimensionless initial and boundaries can be defined as:

(3)

(4)

(5)

where, Bi is the Biot number, Bi = hfr1/kf. Taking the Laplace transformation with respect to τ for Eq. 2 and using the initial condition (3), we can express the differential equation as:

(6)

where, s is the transformed variable. After neglecting the integrals in the coefficient of exponentials, the Keller and Keller’s solution of Eq. 6 is:

(7)

where, D2 is a constant to be determined by the boundary condition (4). After employing Laplace transformation, the boundary condition (4) can be express as:

(8)

Substituting Eq. 7 into Eq. 8, the constant D2 can be express as:

(9)

Thus, Eq. 7 becomes:

(10)

Applying the contour integral method, the inverse transform of the temperature distribution is:

(11)

Where:

(12)

(13)

The λn s in the above equations can be calculated from the following equation:

(14)

The temperature distribution φ(ξ, τ) is then given by the sum of Eq. 12 and 13 associated with the λn’s determined by Eq. 14. The temperature distribution valid for small values of time can be obtained by applying the methods of Ozisik (1980). It can be expressed as:

(15)

From an examination of Eq. 15, it cab be easily seen that the value of the temperature reaches within one percent of its steady state value when the end of fin (ξ = R) with the following condition holds,

(16)

or express as another form,

(17)

where, λ1 is the first root of Eq. 14. The minimum value of time that is needed to reach the steady state τmin can be obtained from Eq. 17. Defining the non-dimensional heat flux at the fin base as:

(18)

After employing the non-dimensional parameters, the heat flux can be expresses as:

(19)

After applying the temperature distribution φ(ξ, τ) in Eq. 15, the non-dimensional heat flux at the base of spiral fin is given by:

(20)

Case 2: A sinusoidal with unit amplitude in base fluid temperature: The temperature variation of the fluid has been changed as

and the non-dimensional parameters for the temperature distribution and frequency are

With the temperature distribution of the spiral fin now was replaced by the symbol of Γ. Therefore, the governing equation is:

(21)

The boundary condition (4) is expressed as:

(22)

The boundary condition (5) and the initial condition (3) can be expressed as:

(23)

Following the same procedure as in the Case 1 by taking the Laplace Transformation with respect to τ for Eq. 21 and 22, and using the Keller and Keller’s method, the governing differential equation is:

(24)

where, F2 is a constant. After employing the Laplace transform, the boundary condition (22) can be expressed as:

(25)

Taking the differentiation of Eq. 24 with respect to ξ and substituting the result into Eq. 25, the constant F2 can be obtained as:

(26)

Therefore, the temperature distribution in s domain is:

(27)

The inverse transform is obtained by using the contour integral method and taking the inverse of Laplace transform, the result is:

(28)

where the first tem, second term and third term can be respectively expressed as:

(29)

And

(30)

And

(31)

And λn’s in Eq. 30 are the same as the solutions of Eq. 14. Introducing the defined parameters

The first and second terms in the temperature distribution can be combined and expressed as :

(32)

where the A, B, C, D, are defined as the followings:

A = cosωτ A coshaX A cosbX–sinωτ A sinh aX A sinbX
B = cosωτ A coshaX A cosbX+sinωτ A cosh aX A cosbX
C = V A a A sinh a A cosb – V A b A cosh a A sinb + W A cosh a A cosb
C = V A a A sinh a A cosb + V A b A cosh a A sinb + W A cosh a A cosb

and

Therefore, the phase angle of the temperature distribution φt’ is:

(33)

The amplitude of the temperature distribution A’mt is:

(34)

where, Nr and Dr represent the numerator and denominator in Eq. 32, respectively. The temperature response is then given by the sum of Eq. 31 and 32. Equation 31 represents the material transient response temperature distribution immediately after the base is subjected to the oscillating base fluid temperature with unit amplitude. And Eq. 32 represents the steady periodic response temperature distribution to the oscillating base fluid temperature with unit amplitude. Introducing the same definition for the heat flux as in the Case 1, the heat flux is defined as:

(35)

After the Laplace transform, the heat flux is:

(36)

Applying the temperature distribution from the sum of Eq. 31 and 32, the non-dimensional heat flux at the fin base with a sinusoidal fluid temperature of unit amplitude can be obtained as:

(37)

where the phase angle under steady state condition φq’ can be shown as:

(38)

where the symbols P, Q, Z1 and Z2 denote, respectively, as:

P = cosha A cosb A C + sinha A sinb A D
Q = cosha A cosb A C – sinha A sinb A C
Z1 = a A C A sinha A cosb – b A C A cosha A sinb + a A D A cosha A sinb + b A D A sinha A cosb
Z2 = a A C A sinha A cosb + b A C A cosha A sinb – a A D A cosha A sinb + b A D A sinha A cosb

The symbols C and D are defined as before. Meanwhile, the amplitude under steady state conditions A’mq is:

(39)

where, Nrq and Drq represent the numerator and denominator in Eq. 38, respectively. In Eq. 37, the first term represents the steady periodic response base heat flux to the oscillating base fluid temperature with unit amplitude. And the second terms represent the initial transient response base heat flow.

RESULTS AND DISCUSSION

Case 1: A unit step change in base fluid temperature: The temperature distribution of the spiral fin φ obtained from Eq. 15 is plotted in Fig. 2-5 for different values of τ, Bi, N, Pi and R.

The result can be draw from Fig. 2-5 that the temperature distribution φ increase as the time elapsed. Also, the absolute slope of temperature distribution φ of the spiral fin base has a trend of decreasing with an increase of time τ which implies that the output heat flux in base of fin will be decreased with an increase of time τ. This result is due to the increase of internal temperature of the spiral fin when time τ is increasing. Meanwhile, for the greater values of fin pitch Pi, the greater values of temperature distribution φ will be at the same values of Bi, R, τ and N except for very small values of time τ and very large values of N.

It can be seen that the greater the value of Bi, the greater the temperature distribution φ will be at the same values of Pi, N, τ and R by comparing Fig. 2 and 3. The temperature distribution φ is a function of Bi because of the heat flux at fin base transferred from convection is larger for the large values of Bi.

It can be seen that the larger the values of N, the smaller the temperature distribution φ at the same values of Pi, τ, R and Bi in comparing the Fig. 3 and 4. Also, from Fig. 3 and 5, it can be seen that the same trend as R increases, the temperature distribution φ decreases at the same value of Bi τ, Pi and N.

Fig. 2: The temperature distribution for R-2, N = 1 Bi =1.0

Fig. 3: The temperature distribution for R = 2, N = 1 Bi = 10

Fig. 4: The temperature distribution for R = 2, N = 5 Bi = 10

The dimensionless time for reaching the steady state τmin with different parameters of τ, Bi, N, Pi and R of Eq. 17, is plotted in Fig. 6 and 7 and discussed as in the followings. From these two plots, the τmin decreases at the same N due to the increase of heat flux at the fin base for the increase of Bi. When the Bi becomes very large, the boundary condition of the fin base can be treated as –Mφ/∂ξ + Bi φ = Bi of which is approximated as Φ Ò 1. This is just the same as the fin base under the condition of a unit step change in base temperature.

Fig. 5: The temperature distribution for R = 3, N = 1 Bi = 10

Fig. 6: The temperature distribution for R = 2 and unit step input

Fig. 7: The time constant distribution for R = 3 and unit step input

This result can be obtained from fact that τmin is almost the same at R = 2 and R = 3 for the case of Bi is greater than 50. Also, the dimensionless time is needed to reach the steady state τmin for different N showed that τmin decreased as the N increased when the other parameters hold the same.


Fig. 8: The heat flux distribution for R = 2, Pi = 0

Fig. 9: The heat flux distribution for R=2, Pi=0.5

However, the opposite trend is found in the parameter R, as the τmin increased when the R increased. And the τmin increased as Pi increased except for the cases of very large N or Bi. The difference of τmin with different values of Pi decreased with the increase of Bi and gradually reached to zero. It implied that the τmin is not a function of Pi when φ = 1.

The heat flux at the fin base q0* with different parameters of τ, Bi, N, Pi and R of Eq. 20 is plotted in Fig. 8-11. The heat flux at fin base is not affected by the τ when examining the Fig. 8, 9 and 11 for the very small values of dimensionless time τ. It is because that different values of N represented the different values of convective heat transfer coefficient for the same shape and size of the spiral fin. Initially the heat flux transferred into the fin by the fin base was used to increase the internal energy of the spiral fin itself. Therefore, the heat flux q0* had little effect on the fin. This is same situation happened in the case of a unit step change of the temperature in fin base.

Fig. 10: The heat flux distribution for R = 2, N = 1, with different Pi

Fig. 11: The heat flux distribution for R = 3, Pi = 0.5 with different N

Comparing the Fig. 8-11, the heat flux at the beginning can be represented by the equation –Mφ/∂ξ = Bi because the low temperature at fin base when the boundary condition is –Mφ/∂ξ + Bi φ = Bi. Therefore, both the temperature distribution φ and the heat flux of fin base q0* increased as the Bi increased for the same values of R, τ, N, Pi from the plots of Fig. 2 and 3. When the values of Bi is larger than 50, the situation can be simulated as the case of a unit step change in fin base temperature. But the heat flux at fin base q0* for the variation of fluid temperature is smaller than the heat flux obtained from the case of a unit step change in the fine base temperature. Due to the fluid convective resistance, q0* is smaller than the special case. Although, it showed that at the beginning, the heat flux at fin base is not affected by R. However, the heat flux at fin base q0* would increase as the values of N or R increased by keeping the other parameters unchanged. Also, it showed the parameter Pi has the same trend as the parameters of N and R.

Fig. 12: The temperature distribution for amplitude at R = 2, N = 1, Pi = 0

Fig. 13: The temperature distribution for amplitude at R = 2, N = 1, Pi = 0.5

Case 2: A sinusoidal with unit amplitude in base fluid temperature: The temperature response is then given by the sum of Eq. 31 and 32. Equation 31 represents the material transient response temperature immediately after the base is subjected to the oscillating base fluid temperature with unit amplitude. And Eq. 32 represents the steady periodic response temperature to the oscillating base fluid temperature with unit amplitude.

Only the steady term of the temperature distribution is on interested in the present study. The phase angle φt’ and amplitude Amt’ of temperature distribution of the spiral fin Γ obtained from Eq. 33 and 34 are plotted in Fig. 12-18 for different values of ω, Bi, N, Pi and R.

Figure 12-18 showed that the amplitude Amt’ of fin base temperature distribution Γ at low frequency can be approximated as the case 1 of whish the fin base is subjected to a variation of unit step change in fluid temperature.

Fig. 14: The phase angle distribution at R = 2, N = 1, Pi = 0 and 0.5

Fig. 15: The amplitude and phase angle of temperature distribution at R = 2, N = 3, Pi = 0

Also, it showed that the amplitude Amt’ of temperature distribution Γ increased as the Bi increased for the same values of Pi, N, ω and R. But the amplitude Amt’ of temperature distribution Γ decreased as the N increased. Also the amplitude Amt’ of temperature distribution Γ increased as the Pi increased except for the very large values of N.


Fig. 16: The amplitude and phase angle of temperature distribution at R = 2, N = 3, Pi = 0.5

Because of its larger importance of the convective heat transfer between the spiral fin and the fluid than the shape of fin itself, the influence of N is greater than Pi on the amplitude Amt’ of temperature distribution.

The effect of the frequency can be obtained from Fig. 12-18. The amplitude Amt’ of temperature distribution Γ decreased as the values of ω increased for the same values of Bi, N, Pi and R. Because the Eq. 34 can be expanded in the form of a Fourier series and the largest ω can be decided from the boundary condition. Therefore, the higher order terms can be transacted for the frequency larger than the largest value of ω. And the error can be estimated from the finite terms.

Figure 12-18 also showed that the phase angle φt’ of fin base temperature distribution Γ for the different values of Pi, Bi, N, ω and R. The phase angle φt’ of temperature distribution Γ decreased as the N increased for the same values of Pi, Bi, ω and R.

The heat flux at the fin base qa* with different parameters of ω, Bi, N, Pi and R of is shown in Eq. 37. Only the steady term of the temperature distribution is on interested, however. The phase angle φt’ and amplitude Amt’ of temperature distribution of the spiral fin qa* obtained from Eq. 38 and 39 are plotted in Fig. 19-22 for different values of ω, Bi, N, Pi and R.

Fig. 17: The amplitude and phase angle of temperature distribution at R = 2, N = 5, Pi = 0

Fig. 18: The amplitude and phase angle of temperature distribution at R = 2, N = 5, Pi = 0.5

Figure 19 and 20 showed that the amplitude Amt’ of the heat flux qa* increased as the input oscillating frequency ω increased except for the very small values of Bi at different values of N, Pi and R.


Fig. 19: The amplitude and phase angle of temperature distribution at R = 2, Pi = 0

Fig. 20: The amplitude and phase angle of temperature distribution at R = 2, Pi = 0.5

Also, the amplitude Amt’ of the heat flux qa* increased as the values of Bi increased. And the amplitude Amt’ of the heat flux qa* increased as the values of N increased except for the small values of ω, for example ω is smaller than 50, too.

Figure 21 showed that the amplitude Amt’ of the heat flux qa* increased as the values of Pi increased for different values of N, Bi, ω and R. Also, Fig. 22 showed that the amplitude Amt’ of the heat flux qa* increased as the values of R increased for different values of N, Bi, ω and Pi except for the very large values of ω.

From above, we can obtain the features that the amplitude Amt’ of the heat flux qa* at any point in the spiral fin of this case is approximately the same as those in the case to a step change in the base temperature for the steady state situation and the low oscillating frequency of which is ω ≤ 1.

Fig. 21: The amplitude and phase angle of temperature distribution at R = 2, N = 1 with different Pi

Fig. 22: The amplitude and phase angle of temperature distribution at, N = 1, Pi = 0.5

However, the greater the values of Pi, the greater the values of the amplitude Amt’ of the heat flux qa* temperature response at the same values of ω, R, Bi and N. Although the deviation the amplitude Amt’ with different values of Pi is less sensitive than those of N, but it also contributes its part in enhancing the heat transfer effect.

CONCLUSIONS

From the case 1, a variation with a unit step change in base heat flux, it can be seen that the large the values of N, the smaller the values of temperature response at the same values of τ, Pi and R. In addition, the greater the values of Pi, the higher the values of temperature response at the same values of N, R and τ. Also it can conclude that the greater the values of R, the smaller the values of temperature response at the same values of τ, N and Pi. It can be seen that τmin decreases with an increase of N at the same values of Pi and R. It also can be seen that the time for reaching steady state, τmin, increases with an increase of R at the same values of N and Pi and Pi at the same values of N and R.

From the case 2, a variation with a sinusoidal base heat flux with unit amplitude e know that the greater the values of N, the smaller the values of temperature response amplitude at the same values of Pi, ω and R; in addition, the higher values of oscillating frequency, the smaller the values of temperature response amplitude at the same values of Pi, R and N. It also can be seen that the greater the values of Pi, the greater the values of temperature response amplitude at the same values of N, ω and R. Also it can conclude that the greater the values of R, the smaller the temperature response amplitude at the same values of N, Pi and ω in a similarity shape of radial dimension in R. However, the temperature response phase angle decreases with an increase of N at the same values of Pi, ω and R. The temperature response angle also increases with an increase of ω for keeping the other conditions in the same values.

In this study, the exact transient solution for unit step input and sinusoidal base heat flux has been obtained. From the results of the above cases, it can be found that the temperature response at the fin base will be influenced by the frequency. Also the amplitude of the input temperature has the direct impact on the time constant for both cases. However, the dominant parameters of Pi, R and N would be the major factors in represented the response of the heat transfer from the base to the spiral fin. These results can be used as the foundation in applying the spiral fin on the industry.

ACKNOWLEDGMENT

The current authors gratefully acknowledge the financial support provided to this study by the National Science Council of Taiwan under Grant No. NSC 95-2221-E-167-028 and NSC 95-2622-E-167-010-CC3.

NOMENCLATURE

A’mq = Dimensionless base heat flow amplitude under steady state
A’mt = Dimensionless temperature amplitude
Bi = hfr1/kf = Biot number
c = Specific heat for material of fin
h = Convective heat transfer coefficient surronding fin
hf = Base convective heat transfer coefficient
k = Thermal conductivity for material of fin
kf = Thermal conductivity of fluid for fin base
N = [hr12/kδ]1/2
p = Pitch of spiral fin
pi = Dimensionless of pitch [p/2πr1]
qb = Heat flow at base of fin
qb* = Dimensionless base heat flow [qbr1/k(4δπr1)(Tf–T)]
qa* = Dimensionless base heat flow Eq. 31
r = Radius of concerned fin
r1 = Inner radius of fin
r2 = Outer radius of fin
R = r2/r1=Dimensionless radius
t = Time
T = Temperature of fin
TA = Temperature parameter
Tf = Base fluid temperature
T0 = Temperature parameter
T = Temperature of fluid surrounding fin
X = Dimensionless radial parameter [(R–ξ)/(1–1)]
α = Thermal diffusivity [k/ρc]
δ = Half thickness of uniform fin
ρ = Density for material of fin
φ = Dimensionless temperature [(T–T∞)/Tf–T)]
Γ = Dimensionless temperature [(T–T)/TA)]
Γ0 = Dimensionless temperature parameter [(T–T)/TA)]
τ = Dimensionless time
ξ = Dimensionless radius
ω = Dimensionless frequency of oscillation [ωr12/α]
j = Frequency of oscillation
φq = Base heat flow phase angle under steady state
φtN = Temperature phase angle
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