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 pinfin 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 fantoheat 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 finandtube 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 onedimensional and twodimensional 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
twodimensional straight fins and circular fins, onedimensional 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 ChaoKuang 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 onedimensional analysis is valid. Onedimensional
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 temperaturedependent 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 temperaturedependent
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 finefficiency curves for
different type of fins.
Because fins are frequently used under unsteady conditions, it is worth studying the transient behavior of fins with timedependent 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 temperaturedependent 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 r_{1}, outer radius r_{2}, pitch P and thermal conductivity k is shown in Fig. 1. The end of the fin, i.e., r = r_{2}, is assumed to be perfectly insulated. It is also assumed that onedimensional 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 T_{f} or subjected to heat flux q_{0}* and from then on, the spiral fin dissipated heat by convection to the environment through a convective heat transfer coefficient h, h_{f} 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 = r_{1}, 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:
where, α = k/ρc is the thermal diffusivity and Biot number is B_{i} = h_{f}r_{1}/k_{f}.
After introducing the nondimensional variables,
The dimensionless governing equation is:
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:
where, B_{i} is the Biot number, B_{i} = h_{f}r_{1}/k_{f}. Taking the Laplace transformation with respect to τ for Eq. 2 and using the initial condition (3), we can express the differential equation as:
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:
where, D_{2} is a constant to be determined by the boundary condition (4). After employing Laplace transformation, the boundary condition (4) can be express as:
Substituting Eq. 7 into Eq. 8, the constant D_{2} can be express as:
Thus, Eq. 7 becomes:
Applying the contour integral method, the inverse transform of the temperature distribution is:
Where:
The λ_{n} s in the above equations can be calculated from the following equation:
The temperature distribution φ(ξ, τ) is then given by the sum
of Eq. 12 and 13 associated with the λ_{n}’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:
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,
or express as another form,
where, λ_{1} is the first root of Eq. 14. The minimum value of time that is needed to reach the steady state τ_{min} can be obtained from Eq. 17. Defining the nondimensional heat flux at the fin base as:
After employing the nondimensional parameters, the heat flux can be expresses as:
After applying the temperature distribution φ(ξ, τ) in Eq. 15, the nondimensional heat flux at the base of spiral fin is given by:
Case 2: A sinusoidal with unit amplitude in base fluid temperature:
The temperature variation of the fluid has been changed as
and the nondimensional 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:
The boundary condition (4) is expressed as:
The boundary condition (5) and the initial condition (3) can be expressed as:
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:
where, F_{2} is a constant. After employing the Laplace transform, the boundary condition (22) can be expressed as:
Taking the differentiation of Eq. 24 with respect to ξ and substituting the result into Eq. 25, the constant F_{2} can be obtained as:
Therefore, the temperature distribution in s domain is:
The inverse transform is obtained by using the contour integral method and taking the inverse of Laplace transform, the result is:
where the first tem, second term and third term can be respectively expressed as:
And
And
And λ_{n}’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 :
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 φ_{t}’ is:
The amplitude of the temperature distribution A’_{mt} is:
where, N_{r} and D_{r} 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:
After the Laplace transform, the heat flux is:
Applying the temperature distribution from the sum of Eq. 31
and 32, the nondimensional heat flux at the fin base with
a sinusoidal fluid temperature of unit amplitude can be obtained as:
where the phase angle under steady state condition φ_{q}’ can be shown as:
where the symbols P, Q, Z_{1} and Z_{2} 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 
Z_{1} 
= 
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 
Z_{2} 
= 
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:
where, N_{rq} and D_{rq} 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. 25 for different values
of τ, B_{i}, N, P_{i} and R.
The result can be draw from Fig. 25 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
P_{i}, the greater values of temperature distribution φ will be
at the same values of B_{i}, 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 B_{i}, the greater the temperature distribution φ will be at the same values of P_{i}, N, τ and R by comparing Fig. 2 and 3. The temperature distribution φ is a function of B_{i} because of the heat flux at fin base transferred from convection is larger for the large values of B_{i}.
It can be seen that the larger the values of N, the smaller the temperature distribution φ at the same values of P_{i}, τ, R and B_{i} 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 B_{i} τ, P_{i} and N.

Fig. 2: 
The temperature distribution for R2, N = 1 B_{i}
=1.0 

Fig. 3: 
The temperature distribution for R = 2, N = 1 B_{i}
= 10 

Fig. 4: 
The temperature distribution for R = 2, N = 5 B_{i}
= 10 
The dimensionless time for reaching the steady state τ_{min} with
different parameters of τ, B_{i}, N, P_{i} and R of Eq.
17, is plotted in Fig. 6 and 7 and discussed
as in the followings. From these two plots, the τ_{min} decreases
at the same N due to the increase of heat flux at the fin base for the increase
of B_{i}. When the B_{i} becomes very large, the boundary condition
of the fin base can be treated as –Mφ/∂ξ + B_{i}
φ = B_{i} 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 B_{i}
= 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 τ_{min} is almost the
same at R = 2 and R = 3 for the case of B_{i} is greater than 50. Also,
the dimensionless time is needed to reach the steady state τ_{min}
for different N showed that τ_{min} decreased as the N increased
when the other parameters hold the same.

Fig. 8: 
The heat flux distribution for R = 2, P_{i} = 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 τ_{min}
increased when the R increased. And the τ_{min} increased as P_{i}
increased except for the cases of very large N or B_{i}. The difference
of τ_{min} with different values of P_{i} decreased with
the increase of B_{i} and gradually reached to zero. It implied that
the τ_{min} is not a function of P_{i} when φ = 1.
The heat flux at the fin base q_{0}* with different parameters of τ,
B_{i}, N, P_{i} and R of Eq. 20 is plotted
in Fig. 811. 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 q_{0}* 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
P_{i} 

Fig. 11: 
The heat flux distribution for R = 3, P_{i} = 0.5
with different N 
Comparing the Fig. 811, the heat flux
at the beginning can be represented by the equation –Mφ/∂ξ
= B_{i} because the low temperature at fin base when the boundary condition
is –Mφ/∂ξ + B_{i} φ = B_{i}. Therefore,
both the temperature distribution φ and the heat flux of fin base q_{0}*
increased as the B_{i} increased for the same values of R, τ, N,
P_{i} from the plots of Fig. 2 and 3.
When the values of B_{i} 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 q_{0}* 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, q_{0}* 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
q_{0}* would increase as the values of N or R increased by keeping the
other parameters unchanged. Also, it showed the parameter P_{i} has
the same trend as the parameters of N and R.

Fig. 12: 
The temperature distribution for amplitude at R = 2, N = 1,
P_{i} = 0 

Fig. 13: 
The temperature distribution for amplitude at R = 2, N = 1,
P_{i} = 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 φ_{t}’ and amplitude A_{mt}’
of temperature distribution of the spiral fin Γ obtained from Eq.
33 and 34 are plotted in Fig. 1218
for different values of ω, B_{i}, N, P_{i} and R.
Figure 1218 showed that the amplitude
A_{mt}’ 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, P_{i}
= 0 and 0.5 

Fig. 15: 
The amplitude and phase angle of temperature distribution
at R = 2, N = 3, P_{i} = 0 
Also, it showed that the amplitude A_{mt}’ of temperature distribution
Γ increased as the B_{i} increased for the same values of P_{i},
N, ω and R. But the amplitude A_{mt}’ of temperature distribution
Γ decreased as the N increased. Also the amplitude A_{mt}’
of temperature distribution Γ increased as the P_{i} increased
except for the very large values of N.

Fig. 16: 
The amplitude and phase angle of temperature distribution
at R = 2, N = 3, P_{i} = 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 P_{i} on the amplitude A_{mt}’ of temperature
distribution.
The effect of the frequency can be obtained from Fig. 1218.
The amplitude A_{mt}’ of temperature distribution Γ decreased
as the values of ω increased for the same values of B_{i}, N, P_{i}
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 1218 also showed that the phase
angle φ_{t}’ of fin base temperature distribution Γ for
the different values of P_{i}, Bi, N, ω and R. The phase angle
φ_{t}’ of temperature distribution Γ decreased as the
N increased for the same values of P_{i}, B_{i}, ω and
R.
The heat flux at the fin base q_{a}* with different parameters of ω,
B_{i}, N, P_{i} and R of is shown in Eq. 37.
Only the steady term of the temperature distribution is on interested, however.
The phase angle φ_{t}’ and amplitude A_{mt}’
of temperature distribution of the spiral fin q_{a}* obtained from Eq.
38 and 39 are plotted in Fig. 1922
for different values of ω, B_{i}, N, P_{i} and R.

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

Fig. 18: 
The amplitude and phase angle of temperature distribution
at R = 2, N = 5, P_{i} = 0.5 
Figure 19 and 20 showed that the amplitude
A_{mt}’ of the heat flux q_{a}* increased as the input
oscillating frequency ω increased except for the very small values of B_{i}
at different values of N, P_{i} and R.

Fig. 19: 
The amplitude and phase angle of temperature distribution
at R = 2, P_{i} = 0 

Fig. 20: 
The amplitude and phase angle of temperature distribution
at R = 2, P_{i} = 0.5 
Also, the amplitude A_{mt}’ of the heat flux q_{a}* increased
as the values of B_{i} increased. And the amplitude A_{mt}’
of the heat flux q_{a}* 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 A_{mt}’ of the heat flux q_{a}* increased as the values of P_{i} increased for different values of N, B_{i}, ω and R. Also, Fig. 22 showed that the amplitude A_{mt}’ of the heat flux q_{a}* increased as the values of R increased for different values of N, B_{i}, ω and P_{i} except for the very large values of ω.
From above, we can obtain the features that the amplitude A_{mt}’
of the heat flux q_{a}* 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 P_{i} 

Fig. 22: 
The amplitude and phase angle of temperature distribution
at, N = 1, P_{i} = 0.5 
However, the greater the values of P_{i}, the greater the values of
the amplitude A_{mt}’ of the heat flux q_{a}* temperature
response at the same values of ω, R, B_{i} and N. Although the
deviation the amplitude A_{mt}’ with different values of P_{i}
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, P_{i} and R. In addition, the greater the values of P_{i}, 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 P_{i}. It can be seen that τ_{min} decreases with an increase of N at the same values of P_{i} and R. It also can be seen that the time for reaching steady state, τ_{min}, increases with an increase of R at the same values of N and P_{i} and P_{i} 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 P_{i}, ω and R; in addition, the higher values of oscillating frequency, the smaller the values of temperature response amplitude at the same values of P_{i}, R and N. It also can be seen that the greater the values of P_{i}, 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, P_{i} 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 P_{i}, 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 P_{i}, 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 952221E167028 and NSC 952622E167010CC3.
NOMENCLATURE
A’_{mq} 
= 
Dimensionless base heat flow amplitude under steady state 
A’_{mt} 
= 
Dimensionless temperature amplitude 
B_{i} = h_{f}r_{1}/k_{f} 
= 
Biot number 
c 
= 
Specific heat for material of fin 
h 
= 
Convective heat transfer coefficient surronding fin 
h_{f} 
= 
Base convective heat transfer coefficient 
k 
= 
Thermal conductivity for material of fin 
k_{f} 
= 
Thermal conductivity of fluid for fin base 
N 
= 
[hr_{1}^{2}/kδ]^{1/2} 
p 
= 
Pitch of spiral fin 
p_{i} 
= 
Dimensionless of pitch [p/2πr_{1}] 
q_{b} 
= 
Heat flow at base of fin 
q_{b}* 
= 
Dimensionless base heat flow [q_{b}r_{1}/k(4δπr_{1})(T_{f}–T_{∞})] 
q_{a}* 
= 
Dimensionless base heat flow Eq. 31 
r 
= 
Radius of concerned fin 
r_{1} 
= 
Inner radius of fin 
r_{2} 
= 
Outer radius of fin 
R 
= 
r_{2}/r_{1}=Dimensionless radius 
t 
= 
Time 
T 
= 
Temperature of fin 
T_{A} 
= 
Temperature parameter 
T_{f} 
= 
Base fluid temperature 
T_{0} 
= 
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∞)/T_{f}–T_{∞})] 
Γ 
= 
Dimensionless temperature [(T–T_{∞})/T_{A})] 
Γ_{0} 
= 
Dimensionless temperature parameter [(T–T_{∞})/T_{A})] 
τ 
= 
Dimensionless time 
ξ 
= 
Dimensionless radius 
ω 
= 
Dimensionless frequency of oscillation [ωr_{1}^{2}/α] 
j 
= 
Frequency of oscillation 
φ_{q}’ 
= 
Base heat flow phase angle under steady state 
φ_{t}N 
= 
Temperature phase angle 