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Mean action time as a measure for fin performance in one dimensional fins of exponential profiles
Applied Mathematics and Computation 238 (2014) 319–328
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Mean action time as a measure for fin performance in one dimensional fins of exponential profiles I. Rusagara, C. Harley ⇑ Centre for Differential Equations, Continuum Mechanics and Applications, School of Computational and Applied Mathematics, University of the Witwatersrand, Private Bag 3, Wits 2050, Johannesburg, South Africa
a r t i c l e
i n f o
Keywords: Numerical relaxation scheme Non-linear heat transfer Thermal conductivity parameter Mean action time Exponential fin profiles
a b s t r a c t The aim of this paper is to numerically solve the one dimensional time dependent heat transfer equation while considering different types of exponential profiles. We consider non-linear thermal conductivity via the power law and employ a numerical relaxation scheme due to its simplicity. Furthermore, we obtain the mean action time and propose it as a novel means of analysing the fin performance. Our results are validated through comparing with benchmark results and our new index for fin performance is shown to provide meaningful insight. Ó 2014 Elsevier Inc. All rights reserved.
1. Introduction There exists a wide range of applications for extended surfaces called fins as has been discussed by several researchers [1–4]. When one introduces notions such as a fin’s profile, form, orientation, height, length and spacing in an array [5–10] into a model for heat transfer one finds that the equation becomes highly non-linear and stiff. In Rusagara and Harley [11] we considered singular profiles and presented an effective computational method for the solution of a similar model to the one considered here. In this research, while our equation remains highly non-linear and stiff, we instead focus on parametric fins with exponential profiles where the thermal conductivity is described by the power law. This choice for the profile is considered in a form which introduces a parameter allowing for a reduction to the rectangular profile. This is done to allow for a comparison of our numerical solutions to analytical solutions. The analytical solutions of equations of the kind as considered in this work are not easily obtained, and may often only be found through a simplification of the model – see [4,11] for concise discussions regarding this and references therein. As such, given that we aim to investigate a highly non-linear partial differential equation, we will employ computational methods for the solution of the equation. There are various methods available for the solution of equations of the type discussed here – see [16–19] and discussions therein particularly pertaining to Spectral numerical integration which is shown to be highly effective. Given the literature – see for instance [13–15] – surrounding numerical relaxation schemes we turn to this methodology for the solution of the nonlinear partial differential equation under discussion. There are various motivations for this choice: (1) the method is able to effectively deal with nonlinear flux and source terms without requiring linearisation, (2) the scheme is simple to implement and (3) is able to achieve higher order accuracy in capturing weak solutions without using Riemann Solvers spatially or systems of algebraic equations temporally.
⇑ Corresponding author. E-mail addresses: [email protected], [email protected] (C. Harley). http://dx.doi.org/10.1016/j.amc.2014.04.013 0096-3003/Ó 2014 Elsevier Inc. All rights reserved.
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Wang et al. [12] stated that a system of highly coupled equations with stiff non-linear source terms needs to be treated in the proper manner computationally so that one does not obtain spurious steady state numerical solutions – this is an appropriate consideration given the nature of the computational method we will employ. As such, we take care when discretising the system to maintain a discrete analogy for the zero relaxation limit which is consistent with the original equation. We consider the one-dimensional case, not due to its simplicity, but rather so that we are able to validate our results through a clear visualisation against benchmark results. An extension to higher dimensions is straight forward given the methodology employed in this work and in [11]. Due to its robustness, as discussed in [14], the scheme shall be applied to the transient heat equation under consideration and be further employed as a means of computing the mean action time for which there is no analytical technique available to us due to the nature of the partial differential equation under discussion. We are interested in the mean action time of the process given that in heat transfer the length of time taken by the process to reach a steady state is of import. In fact, we are interested in introducing a novel means of considering the fin performance, by relating it to the time taken by the process to reach some type of equilibrium. This proposed index is critical since the duration of a process plays a major role in deciding the proportions of a fin so as to be able to accelerate the process of heat transfer. However, as stated by Landman and McGuiness [20], diffusive processes often take an infinite amount of time to come to equilibrium and as such trying to obtain the time at which the process has reached equilibrium may not be possible either computationally (due to often incurred computational error) or analytically (due to the structure of the model). For this reason, we consider an approximation instead, i.e. instead of considering the time taken to reach a steady state we investigate the mean action time as a measure of the time taken to reach some type of equilibrium. The structure of the problem under consideration does not lend itself towards obtaining the mean action time as done by McNabb and Wake [21] and McNabb [22]. In these works an approach is proposed which requires one to employ Poisson’s equation such that the mean action time is easily computed without need of the original problem’s solution [21]. In our case, we are unable to write the partial differential equation in conserved form due to the presence of a complex non-linear flux term and non-linear source term. As such we are unable to follow the work in [21] without certain alterations. In fact, due to this complication and based on the accuracy of relaxation schemes as stated by Jin and Xin [15], we investigate the mean action time via the methodology introduced in McNabb and Wake [21] but by computing the mean action time numerically via a relaxation scheme instead. In order to compute the mean action time numerically in this fashion, given that we require the equation to be in conserved form, we also assume that the thermo-geometric parameter M is small enough so that while it has an impact it does not dominate to the degree that the method of McNabb and Wake [21] is inappropriate. This approach employed is novel and allows us to propose that the mean action time can be used as a means of assessing the fin performance. Our research has two components: firstly, we establish a numerical relaxation scheme and secondly we propose a direct computational approach for computing the mean action time for reaching a steady state for small values of the thermo-geometric parameter M. Given that the mean action times have not as yet been obtained – to the best of the author’s knowledge – by other researchers, we are unable to compare with other established results. We are able to justify our work given the methodology employed and via the results produced by our investigation: we show that the mean action times computed consistently provide 23 of the steady state temperature determined computationally at time smax [20]. The motivation behind this research is the idea that steady state solutions in and of themselves are unable to exhaustively define the fin performance. In fact, we claim that the insight obtained from steady state solutions regarding fin performance is insufficient unless the time (or an approximation thereof) taken by the process to reach the given state can be provided by said solution. 2. Numerical relaxation scheme for one dimensional heat transfer The relaxation scheme is structured via the introduction of a linear system with source term. We consider
@h @ @h M2 hnþ1 ; f ðxÞkðhÞ ¼ @ s @x @x
0 x 1;
s P 0;
ð1Þ
where boundary conditions are as follows
@h ¼ 0 at the fin tip; @xx¼0
ð2Þ
hðs; 1Þ ¼ 1 at the base of the fin:
ð3Þ
The initial condition is given as
hð0; xÞ ¼ 0:
ð4Þ
We define h as the dimensionless temperature, s the dimensionless time, x the dimensionless spatial variable, f ðxÞ the dimensionless fin profile and kðhÞ the dimensionless thermal conductivity. Furthermore, the thermo-geometric parameter 2
bL is defined as M ¼ 2Ph with Ap the fin profile area, P the perimeter of the fin, L the length of the fin, db the fin thickness ka d Ap b
at the base of the fin, hb the heat transfer coefficient at the base and ka the thermal conductivity of the fin at the ambient
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temperature. Eq. (1) and the corresponding boundary conditions (2)–(4) are in dimensionless form. For a description on the energy balance equation and the process of non-dimensionalisation leading to the problem under consideration see [11]. The corresponding relaxation system is
@h @ v þ ¼ M2 hnþ1 @ s @x @v @h 1 þ a2 ¼ ðF v Þ @x @ s @h ¼ 0 and hðs; 1Þ ¼ 1 @xx¼0 @ v ¼ Fð0Þ and v ðs; 1Þ ¼ Fð1Þ; @x x¼0
ð5Þ
where F ¼ f ðxÞkðhÞ @h with a the characteristic speed and the relaxation parameter. The key concept of this theory is that @x the relaxation system should reduce to (1) for ¼ 0 and the partial differential operator of the relaxation system is linear and diagonalizable with two characteristics
v ah With this approach, special care should be taken when discretising the system (5) so that there is still a discrete analogy for the zero relaxation limit which is consistent with the original Eq. (1). 2.1. Numerical discretisation In order to discretise the spatial grid we define the points xiþ1 with mesh width Dxi ¼ xiþ1 xi1 as well as the time step 2 2 2 Dsj ¼ sjþ1 sj such that hji denotes the approximation cell average of h in the cell ½xi1 ; xiþ1 at time sj while hjiþ1 is the approx2
imation of h at x ¼ xiþ1 and 2
2
2
s ¼ sj .
2.1.1. Relaxation scheme Using the integral approach and spatial cell averaging of Eq. (5) we get
dhi 1 þ v 1 v i12 ¼ M2 hnþ1 i ds Dxi iþ2 1 dv i 1 þ a2 h 1 hi1 ¼ ðF i v i Þ 2 ds Dx iþ2 Z x 1 i 2 i 1 2 FðhÞdx þ O Dx ¼ F i Dxi x 1 i
ð6Þ
2
as established and similarly defined by Jin and Xin [15]. The integral defined here is calculated according to the cell averaging technique as done in [23]. By employing an upwind scheme, quantities hiþ1 and v iþ1 are easily defined. This is due to the fact that the system (5) has 2 2 two characteristic variables v ah travelling with characteristic speeds a respectively. Hence
ðv þ ahÞiþ1 ¼ ðv þ ahÞi ;
ð7Þ
ðv ahÞiþ1 ¼ ðv ahÞiþ1 :
ð8Þ
2
2
From (7) we get
v iþ
1 2
þ ahiþ1 ¼ v i þ ahi 2
ð9Þ
and from (8) we get
v iþ
1 2
ahiþ1 ¼ v iþ1 ahiþ1 : 2
ð10Þ
By adding (9) with (10) and after simple algebraic manipulation we obtain
v iþ
1 2
¼
1 a ðv i þ v iþ1 Þ ðhiþ1 hi Þ: 2 2
ð11Þ
Similarly, by subtracting (10) from (9) we find
hiþ1 ¼ 2
1 1 ðhi þ hiþ1 Þ ðv iþ1 v i Þ: 2 2a
From Eq. (11) and (12) we obtain the following important expressions
ð12Þ
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v iþ
1 2
1 a ðv iþ1 v i1 Þ ðhiþ1 2hi þ hi1 Þ 2 2
ð13Þ
1 1 ðhiþ1 hi1 Þ ðv iþ1 2v i þ v i1 Þ: 2 2a
ð14Þ
v i1 ¼ 2
and
hiþ1 hi1 ¼ 2
2
By introducing Eqs. (13) and (14) into the system (6) we finally obtain a relaxation scheme as expressed by the system
dhi 1 a ¼ ðv iþ1 v i1 Þ þ ðhiþ1 2hi þ hi1 Þ M2 hnþ1 ; i 2 Dxi 2 Dxi ds dv i a2 a 1 ¼ ðhiþ1 hi1 Þ þ ðv iþ1 2v i þ v i1 Þ þ ðF i v i Þ 2 D xi ds 2Dxi hiþ1 hi1 F i ¼ f ðxi Þkðhi Þ 2 Dxi F0 ¼ 0
ð15Þ
for the one dimensional heat transfer equation under consideration. 2.1.2. Zero relaxation numerical scheme When structuring the relaxation scheme it was required that the numerical discretisation must have a discrete analogy to the zero relaxation limit which should be consistent with the original partial differential equation. In our case, for ! 0, we have v i ! F i and hence the zero relaxation numerical scheme becomes
dhi 1 a ¼ ðF iþ1 F i1 Þ þ ðhiþ1 2hi þ hi1 Þ M2 hnþ1 ; i 2Dxi 2 D xi ds hiþ1 hi1 ; F i ¼ f ðxi Þkðhi Þ 2Dxi F 0 ¼ 0:
ð16Þ
Hence the first order fully discretized scheme can be expressed as
Ds aDs ðF iþ1 F i1 Þ þ ðhiþ1 2hi þ hi1 Þ DsM2 hnþ1 ; i 2 Dxi 2Dxi hiþ1 hi1 ; F i ¼ f ðxi Þkðhi Þ 2 Dxi F 0 ¼ 0:
hjþ1 ¼ hji i
ð17Þ
The above expression provides a relaxed numerical scheme for heat transfer in one dimensional longitudinal fin profile. In this research we will test this scheme via a comparison against analytical solutions of the steady state equation solved by Turkyilmazoglu [24]. 3. Mean action time A common query in heat transfer is the time taken by the process to reach a specific state when initial and limited boundary information is provided. It is more complex to investigate the time at which such a state is achieved when the governing equation is non-linear such as the case considered in this work. However, complications arise not only due to the non-linearity of the equation, but also due to the fact that it is unknown at what temperature the steady state has been reached. In heat transfer this is frequently the case. McNabb and Wake [21] and McNabb [22] suggested that to compute the averaged quantity known as the mean action time may be a more realistic approach to gaining such information. This approach can easily be applied to scenarios with special boundary values providing the final state. In our case however, we do not have such information and as such we approach the problem differently. 3.1. Numerical mean action time Landman and McGuiness [20] explain that because diffusive processes often take an infinite amount of time to come to equilibrium it is much simpler and more convenient to consider an averaged time. Thus, given that we do not know what the final steady state temperature is, the best approach to use for a non-linear heat transfer problem with temperature dependent thermal conductivity is to consider the mean average time instead as a measure for when the process has reached some kind of equilibrium. In order to follow the methodology as given in McNabb and Wake [21] in this fashion, and then compute the mean action time numerically, we in actual fact require Eq. (1) to be in conserved form, which it clearly is not. As such we consider only
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323
cases for which our thermo-geometric parameter M is small, i.e. M M . This allows us to then continue as proposed by McNabb and Wake [21] where the mean action time is defined as follows
smean ¼ Sup
Z
X
1
sJðx; sÞds
ð18Þ
0
with Jðx; sÞ as the density function. In our case, the appropriate density function can be established by using a similar approach with regard to the decay process. We let Jðx; sÞ ¼ cðxÞ @@hs be the density function associated with @@hs. Then, as per the definition of the density function, we have
Z
Z
1
Jðx; sÞds ¼
0
1
cðxÞ
0
@h ds ¼ 1: @s
This means that @h
s Jðx; sÞ ¼ R 1 @@h
@s
0
ds
ð19Þ
:
From Eqs. (18) and (19) we have
A B
smean ¼ Sup ; X
A¼
Z
1
s
0
B¼
Z
1
@h ds; @s
@h ds: @s
0
That is
R
1 @h s ds smean ¼ Sup R01 @h@s : ds X 0 @s
ð20Þ
In our case we don’t know details regarding the final steady state. Furthermore, given the non-linearity of the equation and the presence of a source term which makes the equation quite complex, it is difficult to find an associated Green’s function in order to structure a Poisson equation for the mean action time as proposed by McNabb and Wake [21]. As a consequence, we instead use the results obtained via our relaxation scheme. This approach has been explicitly provided below. Let
@ @h M2 hnþ1 f ðxÞkðhÞ @x @x
Dðx; sÞ ¼ then
AðxÞ ¼
Z
1
sDðx; sÞds;
ð21Þ
Dðx; sÞds:
ð22Þ
0
BðxÞ ¼
Z
1
0
By employing the finite difference method we have
Dðxi ; sÞ ¼
1 dðxiþ1 ; xiþ1 ; xi Þ dðxi ; xi1 ; xi1 Þ M2 hnþ1 ðxi ; sÞ þ OðDxÞ2 2 2 2 D xi
ð23Þ
with
d xi ; xi1 ; xi1 ¼ f xi1 k hðxi1 ; sÞ ðhðxi ; sÞ hðxi1 ; sÞÞ; 2
2
2
where we compute hðxi ; sÞ via our numerical relaxation scheme. Therefore n X
Aðxi Þ ¼ lim
n!1
Bðxi Þ ¼ lim
n X
n!1
sj Dðxi ; sj ÞDsj ;
ð24Þ
Dðxi ; sj ÞDsj
ð25Þ
j¼1
j¼1
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I. Rusagara, C. Harley / Applied Mathematics and Computation 238 (2014) 319–328
and
s ðxi Þ ¼
Aðxi Þ : Bðxi Þ
ð26Þ
As such we define
smean ¼ Sup fs ðxi Þg:
ð27Þ
xi
This constitutes our explicit numerical mean action time. The significance of this computed value will be discussed in the subsequent sections. 4. Parametric exponential fin profiles and model validation We consider Eq. (1) and define the fin profile as a parametric exponential profile and the thermal conductivity as a power law so that
f ðxÞ ¼ eax and
kðhÞ ¼ hm which gives
@h @ @h M2 hnþ1 ; ¼ eax hm @ s @x @x
0 6 x 6 1;
s P 0:
ð28Þ
The boundary conditions and initial condition are defined as before
@h ¼ 0 at the fin tip; @xx¼0
ð29Þ
hðs; 1Þ ¼ 1 at the base of the fin;
ð30Þ
hð0; xÞ ¼ 0:
ð31Þ
In this research we have chosen to consider a fin with an exponential profile which reduces to the rectangular profile when a ¼ 0. This problem has been solved analytically for steady states [24] and as such we use those results to validate our numerical relaxation scheme. While analytical solutions do exist for steady states [24], it is difficult to obtain such solutions for the time dependent case of Eq. (28). Furthermore, assessing the performance of a fin based solely on steady state solutions is limiting. Thus we consider the time dependent equation and the mean action time as defined in the previous section and propose the latter to be used as a means of assessing the time taken for the process to reach some sort of equilibrium; thus we suggest that smean can be used as a measure for the time taken to reach a steady state. As per the work of McNabb and Wake [21], we find that the mean average time is a useful finite measure to be used for comparison with the time taken by a thermal transition process to obtain equilibrium. This shall be discussed in more detail in the next section. 5. Results and discussion In this section we consider the heat flow in one dimensional exponential fins with varying values of the exponential parameter, thermo-geometric parameter and nonlinear thermal conductivity exponent. In much research regarding heat transfer, rectangular fins are often considered when multiple numerical and analytical methods are employed [4,25,26]. In addition, exact analytical solutions are available for the steady state equation of heat transfer in straight fins of exponential shape [24]. Therefore, we consider these fin shapes in the transient case so as to better validate our numerical results. We obtain numerical solutions via the relaxation scheme and then discuss the results. The variation of the exponential parameter is essential as it allows us to consider different profiles, one of which is the rectangular profile when a ¼ 0, allowing for a comparison to analytical steady state solutions. In order to compute steady state solutions from the time dependent model, we allow our numerical solutions to converge over large time to within a certain tolerance. In this manner we are able to compare our numerical solutions to steady state solutions established analytically by Turkyimazoglu [24]. Figs. 1–3 depict analytical steady state solutions ð Þ as well as the time dependent numerical solutions ð Þ at M ¼ 0:25; 0:5; 0:75; 1:00 and 1.50. These Figures show that the numerical relaxation results of the transient heat equations converge within a reasonable error tolerance to the analytical solutions from the steady state heat equations. As such, our numerical results have been validated via the comparison, allowing us to assume that using the same scheme to compute the mean action time is reasonable.
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I. Rusagara, C. Harley / Applied Mathematics and Computation 238 (2014) 319–328 α=0
1 M=0.25
0.95
0.9
M=0.25 M=0.50
0.9
M=0.50
M=0.75
0.85
0.8 M=0.75 0.7
M=1.00
0.8 θ
θ
α=1
1
M=1.00
M=1.50
0.75 0.7
M=1.50 0.6
0.65 0.6
0.5
0.55 0.4 0
0.2
0.4
0.6
0.8
0.5 0
1
0.2
0.4
x
0.6
0.8
1
x
Fig. 1. Temperature distribution for an exponential fin profile where a ¼ 0 (top) and a ¼ 1 (bottom) with n ¼ m ¼ 1=4 obtained via the relaxation scheme ð Þ compared to a steady state solution ð Þ.
α=2
1
1
M=0.25
M=0.25 M=0.50
M=0.50
0.95
α=3
M=0.75
M=1.00
0.95
M=0.75
M=1.00
0.9
M=1.50
M=1.50
0.85
θ
θ
0.9
0.85 0.8 0.8
0.75
0.7 0
0.2
0.4
0.6
0.8
1
0.75 0
0.2
0.4
x
0.6
0.8
1
x
Fig. 2. Temperature distribution for an exponential fin profile where a ¼ 2 (top) and a ¼ 3 (bottom) with n ¼ m ¼ 1=4 obtained via the relaxation scheme ð Þ compared to a steady state solution ð Þ.
5.1. Temperature distribution and fin efficiency In Figs. 1–3 we have compared the analytical solution of the steady state case to our numerical solutions for the rectangular case and the cases where a ¼ 1; 2; 3 and 4. This was used to highlight how the temperature distribution increases with the increase of the parameter a and hence to show that the fin performance, in terms of a high heat distribution, becomes better with an increase in the parameter value. These results regarding the overall temperature distribution being higher for larger values of the fin shape parameter a comply with [24]. For each case, we vary M to better compare our results of the temperature distribution across different values thereof. Given that it has been shown that M is proportional to the fin length – see [27] – this allows us to make inferences regarding the behaviour of the temperature distribution for different fin lengths. We observe that the temperature distribution is shown to be higher across the fin length the shorter the fin which complies with the idea that short fins shall transfer heat quicker than long fins with similar properties as per the work of Moitsheki and Harley [4] and Harley [27]. Furthermore, from Figs. 1–3 it is clear that the temperature distribution decreases as one moves along the fin from the base to the tip. 5.2. Mean action time as a measure of fin performance One of the most studied and important fin characteristics is the fin efficiency defined as
g¼
Z 0
1
hðxÞnþ1 dx:
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I. Rusagara, C. Harley / Applied Mathematics and Computation 238 (2014) 319–328 α=4
1 M=0.25 M=0.50 M=0.75
M=1.00
M=1.50
θ(x)
0.95
0.9
0.85 0
0.2
0.4
0.6
0.8
1
x
Fig. 3. Temperature distribution for an exponential fin profile where a ¼ 4 with n ¼ m ¼ 1=4 obtained via the relaxation scheme ð state solution ð Þ.
Þ compared to a steady
Fig. 4 depicts the fin efficiencies for both a rectangular fin and an exponential fin profile with parameter a ¼ 3. As depicted we find that the exponential fin profiles are more efficient than the rectangular fin profiles. This complies with our results which showed that the higher the exponential parameter the more efficient is the temperature distribution – see Figs. 1–3. The same results have been obtained from steady states solutions by [24] as shown in the graphics provided. Transient solutions are able to provide us with more than just solutions through time however. In fact, we believe that these solutions will be able to provide us with the transition time between two steady states which plays a major role in the evaluation of the fin performance. Thus far the fin performance is mostly described by the fin tip temperature rise, the fin efficiency and the base heat transfer rate [24] none of which consider the time taken by the temperature distribution process to reach some type of equilibrium. Even if it is difficult to know exactly how long a process may take to reach an equilibrium, the mean action time is able to provide us with an approximation of the total time taken as suggested by McNabb and Wake [21] and Landman and McGuiness [20]. Therefore, to better analyse the fin performance, we consider the mean action time as a means of obtaining an averaged transition time. In order to obtain smean , given how well our relaxation scheme performed against well known benchmark results, we use the same numerical scheme as a means of computing the mean action time as described in a previous section. This is done for small values of the thermo-geometric parameter, as per previous discussions, due to the nature of our equation. A computational measure, smax , of our time taken to equilibrium is obtained by fixing the maximum error between the iterants, i.e. we fix the error allowed as a means of defining computational convergence. This is required, given that in actuality, as per Landman and McGuiness [20], ‘‘diffusion processes take an infinite amount of time to come to equilibrium’’. We fix the tip temperature error allowed between the solutions as ¼ 5 103 . In order to compare smean and smax we employ the
Exponential Rectangular
1
η:Efficiency
0.8
0.6
0.4
0.2
0
1
2
3
4
5
M
6
7
8
9
10
Fig. 4. Comparison of the fin efficiency ðgÞ between a rectangular ða ¼ 0Þ and exponential ða ¼ 3Þ fin profile for n ¼ m ¼ 1=4 and varying M.
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I. Rusagara, C. Harley / Applied Mathematics and Computation 238 (2014) 319–328 Table 1
smean against M for a ¼ 0 and n ¼ m ¼ 1=4. M hs
mean
ð0Þ
hexact ð0Þ mean
%
s smax
0.25
0.50
0.75
1.00
1.50
66.94
66.91
66.80
66.61
65.88
0.6016 2.2784
0.5701 2.1306
0.5267 1.9149
0.4797 1.6677
0.3944 1.2039
Table 2
smean against M for a ¼ 1 and n ¼ m ¼ 1=4. M hs
mean
ð0Þ
hexact ð0Þ mean
%
s smax
0.25
0.50
0.75
1.00
1.50
66.95
66.92
66.86
66.77
66.44
0.3217 1.1856
0.3131 1.1461
0.3001 1.0847
0.28447 1.007
0.2507 0.8292
Table 3
smean against M for a ¼ 2 and n ¼ m ¼ 1=4. M hs
mean
ð0Þ
hexact ð0Þ
%
smean smax
0.25
0.50
0.75
1.00
1.50
66.90
66.88
66.82
66.76
66.52
0.1812 0.6492
0.1786 0.6353
0.1744 0.6133
0.1691 0.5847
0.1563 0.5155
temperature at the tip; this choice was provided by Turkyilmazoglou [24] where the tip temperature was classified among the most important characteristics of fins studied in engineering heat transfer problems. In many cases, our numerical results were taking extremely long to converge and as such the time taken to stabilize could not be determined with the hardware available to us. This again is an indication of the difficulties involved in finding the time taken for equilibrium to be reached. Our intention is to propose the mean action time as an additional performance index based on the duration of the process. Tables 1–3 show the mean action time for different types of exponential fin profiles. Table 1 is for a ¼ 0 (rectangular case), Table 2 for a ¼ 1 and Table 3 for a ¼ 2. For various values of the thermo-geometric parameter we give the ratio of the solution at the mean action time against the exact solution at the tip, the computed mean action time and the maximum time used for convergence of our numerical solution. We find that the highest times taken to approximately converge to the steady state analytical solutions were extremely high when compared to the mean action times that were computed. In fact we find that the tip temperature at the mean action time reaches approximately two thirds 23 of the tip temperature associated with the maximum time taken to convergence. Furthermore, we notice that smax =smean 3:7 (as M increases this ratio decreases slightly) across the various values of the thermo-geometric parameter which again shows that our methodology produces consistent results for small M. As such we find that, as per Landman and McGuiness’s suggestion [20], we have taken a more averaged approach and determined when the average temperature is a fixed fraction of the final equilibrium value, instead of obtaining the final time taken to reach the steady state. However the fact that this fraction is fixed at two thirds means that this information is useful in determining the time taken for the process to reach equilibrium and hence acts as a means of assessing the performance of the fin in terms of the length of time taken by this process. Furthermore, Tables 1–3 show that the mean action time taken to reach a steady state is less for the exponential profiles than for the rectangular case. It is seen that the higher the values of the exponential parameter a, the lower are the corresponding mean action times. This means that for applications that require the process to reach equilibrium fast it would be better to use exponential fin profiles with a larger exponential coefficient parameter. Therefore, one may select an appropriate profile which fulfills the requirement for reaching equilibrium at a certain speed; a useful performance indicator which can assist in this regard is the mean action time. 6. Conclusions The work was subdivided into two main parts: (1) establishing a relaxation system and numerically solving the corresponding system of equations and, (2) computing the mean action time to steady states and using this as an index for assessing the performance of the fin in terms of the time taken to reach some type of equilibrium. In our study, we have used a numerical relaxation scheme to solve a time dependent one dimensional heat transfer equation with different exponential fin profiles. The numerical relaxation scheme was implemented due to its simplicity and accuracy as described in the literature by Jin and Xin [15]. To the best of the authors’ knowledge this is the first time that this method has been employed within the context of such a problem. The results obtained complied with steady state
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analytical solutions [24] upon computational convergence. As such we were able to use this scheme to further investigate the mean action time of the process. Results obtained via the established numerical relaxation scheme showed that exponential fin profiles are more efficient in terms of heat transfer than the rectangular profile. The analysis of the mean action times obtained confirmed Landman and McGuiness’s suggestion that diffusive processes take a considerable amount of time to come to equilibrium [20]. We were able to show that the methodology we employed provided us with a fixed fraction of the final equilibrium temperature reached which becomes useful in assessing the performance of the fin profiles considered. Considering the mean action time along with the usually determined fin efficiency when evaluating the fin performance allows us to view the efficiency of a fin in a multi-dimensional fashion. Acknowledgement CH acknowledges support from the National Research Foundation, South Africa, under Grant No. 79184. The statements made and views expressed are, however, solely the responsibility of the author. References [1] V.D. Rao, S.V. Naidu, B.G. Rao, K.V. Sharma, Heat transfer from a horizontal fin array by natural convection and radiation – a conjugate analysis, Int. J. Heat Mass Transfer 49 (2006) 3379–3391. [2] E.M. Sparrow, S.B. Vemuri, Natural convection–radiation heat transfer from highly populated pin-fin arrays, ASME J. Heat Transfer 107 (1985) 190–197. [3] A.D. Kraus, A. Aziz, J. Welty, Extended Surface Heat Transfer, Wiley: Interscience Publication, USA, 2001. [4] R.J. Moitsheki, C. Harley, Transient heat transfer in longitudinal fins of various profiles with temperature-dependent thermal conductivity and heat transfer coefficient, Pranama-J. Phys. 77 (2011) 519–532. [5] A. Guvenc, H. Yancu, An experiment investigation on performance of fins on a horizontal base in a free convection heat transfer, Heat Mass Transfer 37 (2001) 409–416. [6] A.I. Zografos, J.E. Sunderland, Natural convection from pin fin arrays, Exp. Therm. Fluid Sci. 3 (1990) 440–449. [7] E.M. Sparrow, S.B. Vemuri, Orientation effects on natural convection or radiation pin-fin arrays, Int. J. Heat Mass Transfer 29 (1986) 359–368. [8] G. Guglielmini, E. Nannel, G. Tanda, Natural convection and radiation heat transfer from staggered vertical fins, Int. J. Heat Mass Transfer 30 (1987) 1941–1948. [9] H. Yuncu, G. Anbar, An experimental investigation on performance of fins on a horizontal base in free convection heat transfer, Heat Mass Transfer 33 (1998) 507–514. [10] R. Rao, S.P. Venkateshan, Experimental study of free convection and radiation in horizontal fin arrays, Int. J. Heat Mass Transfer 39 (1996) 779–789. [11] I. Rusagara, C. Harley, A numerical well-balanced scheme for one dimensional heat transfer in longitudinal triangular fins, Math. Prob. Eng. (2013), http://dx.doi.org/10.1155/2013/609536. [12] W. Wang, C.W. Shu, H.C. Yee, B. Sjogreen, On Well-balanced Schemes for Non-equilibrium Flow with Stiff Source Terms, Center for Turbulence Research, Annual Research Briefs, 2008. [13] A.I. Delis, Th. Katsaounis, Numerical solution of two-dimensional shallow water equations by application of relaxation methods, Appl. Math. Model. 29 (2005) 754–783. [14] A.I. Delis, Th. Katsaounis, Relaxation scheme for shallow water equations, Int. J. Numer. Methods Fluids 41 (2003) 695–719. [15] S. Jin, Z. Xin, The relaxation schemes for systems of conservation laws in arbitrary space dimensions, Commun. Pure Appl. Math. 48 (1995) 235–276. [16] M. Turkyilmazoglu, Unsteady mhd flow with variable viscosity: applications of spectral scheme, Int. J. Therm. Sci. 49 (3) (2010) 563–570. [17] M. Turkyilmazoglu, Numerical computation of unsteady flows with an implicit spectral method over a disk in still air, Numer. Heat Transfer A-Appl. 57 (1) (2010) 40–53. [18] M. Turkyilmazoglu, MHD fluid flow and heat transfer due to a shrinking rotating disk, Comput. Fluids 90 (2014) 51–56. [19] M. Turkyilmazoglu, MHD fluid flow and heat transfer due to a stretching rotating disk, Int. J. Therm. Sci. 51 (1) (2012) 195–201. [20] K. Landman, M. McGuiness, Mean action time for diffusive process, J. Appl. Math. Decis. Sci. 4 (2) (2000) 125–141. [21] A. McNabb, G.C. Wake, Heat conduction and finite measures for transition between steady states, IMA J. Appl. Math. 47 (1991) 193–206. [22] A. McNabb, Means action times, time lags, and mean first passage times for some diffusion problems, Math. Comput. Model. 18 (10) (1993) 123–129. [23] R.J. LeVeque, Numerical Methods for Conservation Laws, Birkhäuser Verlag, Berlin, Germany, 1992. [24] M. Turkyilmazoglu, Exact solutions to heat transfer in straight fins of varying exponential shape having temperature dependent properties, Int. J. Therm. Sci. 55 (2012) 69–79. [25] S. Kim, C.H. Huang, A series solution of the non-linear fin problem with temperature-dependent thermal conductivity and heat transfer coefficient, J. Phys. D Appl. Phys. 40 (2007) 2979–2987. [26] R.J. Moitsheki, T. Hayat, M.Y. Malik, Some exact solutions of the fin problem with a power law temperature-dependent thermal conductivity, Nonlinear Anal. Real World Appl. 11 (2010) 3287–3294. [27] C. Harley, Asymptotic and dynamical analyses of heat transfer through a rectangular longitudinal fin, J. Appl. Math. 2013 (2013) 8 pages (Article ID 987327).
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Mean action time as a measure for fin performance in one dimensional fins of exponential profiles I. Rusagara, C. Harley ⇑ Centre for Differential Equations, Continuum Mechanics and Applications, School of Computational and Applied Mathematics, University of the Witwatersrand, Private Bag 3, Wits 2050, Johannesburg, South Africa
a r t i c l e
i n f o
Keywords: Numerical relaxation scheme Non-linear heat transfer Thermal conductivity parameter Mean action time Exponential fin profiles
a b s t r a c t The aim of this paper is to numerically solve the one dimensional time dependent heat transfer equation while considering different types of exponential profiles. We consider non-linear thermal conductivity via the power law and employ a numerical relaxation scheme due to its simplicity. Furthermore, we obtain the mean action time and propose it as a novel means of analysing the fin performance. Our results are validated through comparing with benchmark results and our new index for fin performance is shown to provide meaningful insight. Ó 2014 Elsevier Inc. All rights reserved.
1. Introduction There exists a wide range of applications for extended surfaces called fins as has been discussed by several researchers [1–4]. When one introduces notions such as a fin’s profile, form, orientation, height, length and spacing in an array [5–10] into a model for heat transfer one finds that the equation becomes highly non-linear and stiff. In Rusagara and Harley [11] we considered singular profiles and presented an effective computational method for the solution of a similar model to the one considered here. In this research, while our equation remains highly non-linear and stiff, we instead focus on parametric fins with exponential profiles where the thermal conductivity is described by the power law. This choice for the profile is considered in a form which introduces a parameter allowing for a reduction to the rectangular profile. This is done to allow for a comparison of our numerical solutions to analytical solutions. The analytical solutions of equations of the kind as considered in this work are not easily obtained, and may often only be found through a simplification of the model – see [4,11] for concise discussions regarding this and references therein. As such, given that we aim to investigate a highly non-linear partial differential equation, we will employ computational methods for the solution of the equation. There are various methods available for the solution of equations of the type discussed here – see [16–19] and discussions therein particularly pertaining to Spectral numerical integration which is shown to be highly effective. Given the literature – see for instance [13–15] – surrounding numerical relaxation schemes we turn to this methodology for the solution of the nonlinear partial differential equation under discussion. There are various motivations for this choice: (1) the method is able to effectively deal with nonlinear flux and source terms without requiring linearisation, (2) the scheme is simple to implement and (3) is able to achieve higher order accuracy in capturing weak solutions without using Riemann Solvers spatially or systems of algebraic equations temporally.
⇑ Corresponding author. E-mail addresses: [email protected], [email protected] (C. Harley). http://dx.doi.org/10.1016/j.amc.2014.04.013 0096-3003/Ó 2014 Elsevier Inc. All rights reserved.
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Wang et al. [12] stated that a system of highly coupled equations with stiff non-linear source terms needs to be treated in the proper manner computationally so that one does not obtain spurious steady state numerical solutions – this is an appropriate consideration given the nature of the computational method we will employ. As such, we take care when discretising the system to maintain a discrete analogy for the zero relaxation limit which is consistent with the original equation. We consider the one-dimensional case, not due to its simplicity, but rather so that we are able to validate our results through a clear visualisation against benchmark results. An extension to higher dimensions is straight forward given the methodology employed in this work and in [11]. Due to its robustness, as discussed in [14], the scheme shall be applied to the transient heat equation under consideration and be further employed as a means of computing the mean action time for which there is no analytical technique available to us due to the nature of the partial differential equation under discussion. We are interested in the mean action time of the process given that in heat transfer the length of time taken by the process to reach a steady state is of import. In fact, we are interested in introducing a novel means of considering the fin performance, by relating it to the time taken by the process to reach some type of equilibrium. This proposed index is critical since the duration of a process plays a major role in deciding the proportions of a fin so as to be able to accelerate the process of heat transfer. However, as stated by Landman and McGuiness [20], diffusive processes often take an infinite amount of time to come to equilibrium and as such trying to obtain the time at which the process has reached equilibrium may not be possible either computationally (due to often incurred computational error) or analytically (due to the structure of the model). For this reason, we consider an approximation instead, i.e. instead of considering the time taken to reach a steady state we investigate the mean action time as a measure of the time taken to reach some type of equilibrium. The structure of the problem under consideration does not lend itself towards obtaining the mean action time as done by McNabb and Wake [21] and McNabb [22]. In these works an approach is proposed which requires one to employ Poisson’s equation such that the mean action time is easily computed without need of the original problem’s solution [21]. In our case, we are unable to write the partial differential equation in conserved form due to the presence of a complex non-linear flux term and non-linear source term. As such we are unable to follow the work in [21] without certain alterations. In fact, due to this complication and based on the accuracy of relaxation schemes as stated by Jin and Xin [15], we investigate the mean action time via the methodology introduced in McNabb and Wake [21] but by computing the mean action time numerically via a relaxation scheme instead. In order to compute the mean action time numerically in this fashion, given that we require the equation to be in conserved form, we also assume that the thermo-geometric parameter M is small enough so that while it has an impact it does not dominate to the degree that the method of McNabb and Wake [21] is inappropriate. This approach employed is novel and allows us to propose that the mean action time can be used as a means of assessing the fin performance. Our research has two components: firstly, we establish a numerical relaxation scheme and secondly we propose a direct computational approach for computing the mean action time for reaching a steady state for small values of the thermo-geometric parameter M. Given that the mean action times have not as yet been obtained – to the best of the author’s knowledge – by other researchers, we are unable to compare with other established results. We are able to justify our work given the methodology employed and via the results produced by our investigation: we show that the mean action times computed consistently provide 23 of the steady state temperature determined computationally at time smax [20]. The motivation behind this research is the idea that steady state solutions in and of themselves are unable to exhaustively define the fin performance. In fact, we claim that the insight obtained from steady state solutions regarding fin performance is insufficient unless the time (or an approximation thereof) taken by the process to reach the given state can be provided by said solution. 2. Numerical relaxation scheme for one dimensional heat transfer The relaxation scheme is structured via the introduction of a linear system with source term. We consider
@h @ @h M2 hnþ1 ; f ðxÞkðhÞ ¼ @ s @x @x
0 x 1;
s P 0;
ð1Þ
where boundary conditions are as follows
@h ¼ 0 at the fin tip; @xx¼0
ð2Þ
hðs; 1Þ ¼ 1 at the base of the fin:
ð3Þ
The initial condition is given as
hð0; xÞ ¼ 0:
ð4Þ
We define h as the dimensionless temperature, s the dimensionless time, x the dimensionless spatial variable, f ðxÞ the dimensionless fin profile and kðhÞ the dimensionless thermal conductivity. Furthermore, the thermo-geometric parameter 2
bL is defined as M ¼ 2Ph with Ap the fin profile area, P the perimeter of the fin, L the length of the fin, db the fin thickness ka d Ap b
at the base of the fin, hb the heat transfer coefficient at the base and ka the thermal conductivity of the fin at the ambient
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temperature. Eq. (1) and the corresponding boundary conditions (2)–(4) are in dimensionless form. For a description on the energy balance equation and the process of non-dimensionalisation leading to the problem under consideration see [11]. The corresponding relaxation system is
@h @ v þ ¼ M2 hnþ1 @ s @x @v @h 1 þ a2 ¼ ðF v Þ @x @ s @h ¼ 0 and hðs; 1Þ ¼ 1 @xx¼0 @ v ¼ Fð0Þ and v ðs; 1Þ ¼ Fð1Þ; @x x¼0
ð5Þ
where F ¼ f ðxÞkðhÞ @h with a the characteristic speed and the relaxation parameter. The key concept of this theory is that @x the relaxation system should reduce to (1) for ¼ 0 and the partial differential operator of the relaxation system is linear and diagonalizable with two characteristics
v ah With this approach, special care should be taken when discretising the system (5) so that there is still a discrete analogy for the zero relaxation limit which is consistent with the original Eq. (1). 2.1. Numerical discretisation In order to discretise the spatial grid we define the points xiþ1 with mesh width Dxi ¼ xiþ1 xi1 as well as the time step 2 2 2 Dsj ¼ sjþ1 sj such that hji denotes the approximation cell average of h in the cell ½xi1 ; xiþ1 at time sj while hjiþ1 is the approx2
imation of h at x ¼ xiþ1 and 2
2
2
s ¼ sj .
2.1.1. Relaxation scheme Using the integral approach and spatial cell averaging of Eq. (5) we get
dhi 1 þ v 1 v i12 ¼ M2 hnþ1 i ds Dxi iþ2 1 dv i 1 þ a2 h 1 hi1 ¼ ðF i v i Þ 2 ds Dx iþ2 Z x 1 i 2 i 1 2 FðhÞdx þ O Dx ¼ F i Dxi x 1 i
ð6Þ
2
as established and similarly defined by Jin and Xin [15]. The integral defined here is calculated according to the cell averaging technique as done in [23]. By employing an upwind scheme, quantities hiþ1 and v iþ1 are easily defined. This is due to the fact that the system (5) has 2 2 two characteristic variables v ah travelling with characteristic speeds a respectively. Hence
ðv þ ahÞiþ1 ¼ ðv þ ahÞi ;
ð7Þ
ðv ahÞiþ1 ¼ ðv ahÞiþ1 :
ð8Þ
2
2
From (7) we get
v iþ
1 2
þ ahiþ1 ¼ v i þ ahi 2
ð9Þ
and from (8) we get
v iþ
1 2
ahiþ1 ¼ v iþ1 ahiþ1 : 2
ð10Þ
By adding (9) with (10) and after simple algebraic manipulation we obtain
v iþ
1 2
¼
1 a ðv i þ v iþ1 Þ ðhiþ1 hi Þ: 2 2
ð11Þ
Similarly, by subtracting (10) from (9) we find
hiþ1 ¼ 2
1 1 ðhi þ hiþ1 Þ ðv iþ1 v i Þ: 2 2a
From Eq. (11) and (12) we obtain the following important expressions
ð12Þ
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v iþ
1 2
1 a ðv iþ1 v i1 Þ ðhiþ1 2hi þ hi1 Þ 2 2
ð13Þ
1 1 ðhiþ1 hi1 Þ ðv iþ1 2v i þ v i1 Þ: 2 2a
ð14Þ
v i1 ¼ 2
and
hiþ1 hi1 ¼ 2
2
By introducing Eqs. (13) and (14) into the system (6) we finally obtain a relaxation scheme as expressed by the system
dhi 1 a ¼ ðv iþ1 v i1 Þ þ ðhiþ1 2hi þ hi1 Þ M2 hnþ1 ; i 2 Dxi 2 Dxi ds dv i a2 a 1 ¼ ðhiþ1 hi1 Þ þ ðv iþ1 2v i þ v i1 Þ þ ðF i v i Þ 2 D xi ds 2Dxi hiþ1 hi1 F i ¼ f ðxi Þkðhi Þ 2 Dxi F0 ¼ 0
ð15Þ
for the one dimensional heat transfer equation under consideration. 2.1.2. Zero relaxation numerical scheme When structuring the relaxation scheme it was required that the numerical discretisation must have a discrete analogy to the zero relaxation limit which should be consistent with the original partial differential equation. In our case, for ! 0, we have v i ! F i and hence the zero relaxation numerical scheme becomes
dhi 1 a ¼ ðF iþ1 F i1 Þ þ ðhiþ1 2hi þ hi1 Þ M2 hnþ1 ; i 2Dxi 2 D xi ds hiþ1 hi1 ; F i ¼ f ðxi Þkðhi Þ 2Dxi F 0 ¼ 0:
ð16Þ
Hence the first order fully discretized scheme can be expressed as
Ds aDs ðF iþ1 F i1 Þ þ ðhiþ1 2hi þ hi1 Þ DsM2 hnþ1 ; i 2 Dxi 2Dxi hiþ1 hi1 ; F i ¼ f ðxi Þkðhi Þ 2 Dxi F 0 ¼ 0:
hjþ1 ¼ hji i
ð17Þ
The above expression provides a relaxed numerical scheme for heat transfer in one dimensional longitudinal fin profile. In this research we will test this scheme via a comparison against analytical solutions of the steady state equation solved by Turkyilmazoglu [24]. 3. Mean action time A common query in heat transfer is the time taken by the process to reach a specific state when initial and limited boundary information is provided. It is more complex to investigate the time at which such a state is achieved when the governing equation is non-linear such as the case considered in this work. However, complications arise not only due to the non-linearity of the equation, but also due to the fact that it is unknown at what temperature the steady state has been reached. In heat transfer this is frequently the case. McNabb and Wake [21] and McNabb [22] suggested that to compute the averaged quantity known as the mean action time may be a more realistic approach to gaining such information. This approach can easily be applied to scenarios with special boundary values providing the final state. In our case however, we do not have such information and as such we approach the problem differently. 3.1. Numerical mean action time Landman and McGuiness [20] explain that because diffusive processes often take an infinite amount of time to come to equilibrium it is much simpler and more convenient to consider an averaged time. Thus, given that we do not know what the final steady state temperature is, the best approach to use for a non-linear heat transfer problem with temperature dependent thermal conductivity is to consider the mean average time instead as a measure for when the process has reached some kind of equilibrium. In order to follow the methodology as given in McNabb and Wake [21] in this fashion, and then compute the mean action time numerically, we in actual fact require Eq. (1) to be in conserved form, which it clearly is not. As such we consider only
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323
cases for which our thermo-geometric parameter M is small, i.e. M M . This allows us to then continue as proposed by McNabb and Wake [21] where the mean action time is defined as follows
smean ¼ Sup
Z
X
1
sJðx; sÞds
ð18Þ
0
with Jðx; sÞ as the density function. In our case, the appropriate density function can be established by using a similar approach with regard to the decay process. We let Jðx; sÞ ¼ cðxÞ @@hs be the density function associated with @@hs. Then, as per the definition of the density function, we have
Z
Z
1
Jðx; sÞds ¼
0
1
cðxÞ
0
@h ds ¼ 1: @s
This means that @h
s Jðx; sÞ ¼ R 1 @@h
@s
0
ds
ð19Þ
:
From Eqs. (18) and (19) we have
A B
smean ¼ Sup ; X
A¼
Z
1
s
0
B¼
Z
1
@h ds; @s
@h ds: @s
0
That is
R
1 @h s ds smean ¼ Sup R01 @h@s : ds X 0 @s
ð20Þ
In our case we don’t know details regarding the final steady state. Furthermore, given the non-linearity of the equation and the presence of a source term which makes the equation quite complex, it is difficult to find an associated Green’s function in order to structure a Poisson equation for the mean action time as proposed by McNabb and Wake [21]. As a consequence, we instead use the results obtained via our relaxation scheme. This approach has been explicitly provided below. Let
@ @h M2 hnþ1 f ðxÞkðhÞ @x @x
Dðx; sÞ ¼ then
AðxÞ ¼
Z
1
sDðx; sÞds;
ð21Þ
Dðx; sÞds:
ð22Þ
0
BðxÞ ¼
Z
1
0
By employing the finite difference method we have
Dðxi ; sÞ ¼
1 dðxiþ1 ; xiþ1 ; xi Þ dðxi ; xi1 ; xi1 Þ M2 hnþ1 ðxi ; sÞ þ OðDxÞ2 2 2 2 D xi
ð23Þ
with
d xi ; xi1 ; xi1 ¼ f xi1 k hðxi1 ; sÞ ðhðxi ; sÞ hðxi1 ; sÞÞ; 2
2
2
where we compute hðxi ; sÞ via our numerical relaxation scheme. Therefore n X
Aðxi Þ ¼ lim
n!1
Bðxi Þ ¼ lim
n X
n!1
sj Dðxi ; sj ÞDsj ;
ð24Þ
Dðxi ; sj ÞDsj
ð25Þ
j¼1
j¼1
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and
s ðxi Þ ¼
Aðxi Þ : Bðxi Þ
ð26Þ
As such we define
smean ¼ Sup fs ðxi Þg:
ð27Þ
xi
This constitutes our explicit numerical mean action time. The significance of this computed value will be discussed in the subsequent sections. 4. Parametric exponential fin profiles and model validation We consider Eq. (1) and define the fin profile as a parametric exponential profile and the thermal conductivity as a power law so that
f ðxÞ ¼ eax and
kðhÞ ¼ hm which gives
@h @ @h M2 hnþ1 ; ¼ eax hm @ s @x @x
0 6 x 6 1;
s P 0:
ð28Þ
The boundary conditions and initial condition are defined as before
@h ¼ 0 at the fin tip; @xx¼0
ð29Þ
hðs; 1Þ ¼ 1 at the base of the fin;
ð30Þ
hð0; xÞ ¼ 0:
ð31Þ
In this research we have chosen to consider a fin with an exponential profile which reduces to the rectangular profile when a ¼ 0. This problem has been solved analytically for steady states [24] and as such we use those results to validate our numerical relaxation scheme. While analytical solutions do exist for steady states [24], it is difficult to obtain such solutions for the time dependent case of Eq. (28). Furthermore, assessing the performance of a fin based solely on steady state solutions is limiting. Thus we consider the time dependent equation and the mean action time as defined in the previous section and propose the latter to be used as a means of assessing the time taken for the process to reach some sort of equilibrium; thus we suggest that smean can be used as a measure for the time taken to reach a steady state. As per the work of McNabb and Wake [21], we find that the mean average time is a useful finite measure to be used for comparison with the time taken by a thermal transition process to obtain equilibrium. This shall be discussed in more detail in the next section. 5. Results and discussion In this section we consider the heat flow in one dimensional exponential fins with varying values of the exponential parameter, thermo-geometric parameter and nonlinear thermal conductivity exponent. In much research regarding heat transfer, rectangular fins are often considered when multiple numerical and analytical methods are employed [4,25,26]. In addition, exact analytical solutions are available for the steady state equation of heat transfer in straight fins of exponential shape [24]. Therefore, we consider these fin shapes in the transient case so as to better validate our numerical results. We obtain numerical solutions via the relaxation scheme and then discuss the results. The variation of the exponential parameter is essential as it allows us to consider different profiles, one of which is the rectangular profile when a ¼ 0, allowing for a comparison to analytical steady state solutions. In order to compute steady state solutions from the time dependent model, we allow our numerical solutions to converge over large time to within a certain tolerance. In this manner we are able to compare our numerical solutions to steady state solutions established analytically by Turkyimazoglu [24]. Figs. 1–3 depict analytical steady state solutions ð Þ as well as the time dependent numerical solutions ð Þ at M ¼ 0:25; 0:5; 0:75; 1:00 and 1.50. These Figures show that the numerical relaxation results of the transient heat equations converge within a reasonable error tolerance to the analytical solutions from the steady state heat equations. As such, our numerical results have been validated via the comparison, allowing us to assume that using the same scheme to compute the mean action time is reasonable.
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I. Rusagara, C. Harley / Applied Mathematics and Computation 238 (2014) 319–328 α=0
1 M=0.25
0.95
0.9
M=0.25 M=0.50
0.9
M=0.50
M=0.75
0.85
0.8 M=0.75 0.7
M=1.00
0.8 θ
θ
α=1
1
M=1.00
M=1.50
0.75 0.7
M=1.50 0.6
0.65 0.6
0.5
0.55 0.4 0
0.2
0.4
0.6
0.8
0.5 0
1
0.2
0.4
x
0.6
0.8
1
x
Fig. 1. Temperature distribution for an exponential fin profile where a ¼ 0 (top) and a ¼ 1 (bottom) with n ¼ m ¼ 1=4 obtained via the relaxation scheme ð Þ compared to a steady state solution ð Þ.
α=2
1
1
M=0.25
M=0.25 M=0.50
M=0.50
0.95
α=3
M=0.75
M=1.00
0.95
M=0.75
M=1.00
0.9
M=1.50
M=1.50
0.85
θ
θ
0.9
0.85 0.8 0.8
0.75
0.7 0
0.2
0.4
0.6
0.8
1
0.75 0
0.2
0.4
x
0.6
0.8
1
x
Fig. 2. Temperature distribution for an exponential fin profile where a ¼ 2 (top) and a ¼ 3 (bottom) with n ¼ m ¼ 1=4 obtained via the relaxation scheme ð Þ compared to a steady state solution ð Þ.
5.1. Temperature distribution and fin efficiency In Figs. 1–3 we have compared the analytical solution of the steady state case to our numerical solutions for the rectangular case and the cases where a ¼ 1; 2; 3 and 4. This was used to highlight how the temperature distribution increases with the increase of the parameter a and hence to show that the fin performance, in terms of a high heat distribution, becomes better with an increase in the parameter value. These results regarding the overall temperature distribution being higher for larger values of the fin shape parameter a comply with [24]. For each case, we vary M to better compare our results of the temperature distribution across different values thereof. Given that it has been shown that M is proportional to the fin length – see [27] – this allows us to make inferences regarding the behaviour of the temperature distribution for different fin lengths. We observe that the temperature distribution is shown to be higher across the fin length the shorter the fin which complies with the idea that short fins shall transfer heat quicker than long fins with similar properties as per the work of Moitsheki and Harley [4] and Harley [27]. Furthermore, from Figs. 1–3 it is clear that the temperature distribution decreases as one moves along the fin from the base to the tip. 5.2. Mean action time as a measure of fin performance One of the most studied and important fin characteristics is the fin efficiency defined as
g¼
Z 0
1
hðxÞnþ1 dx:
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I. Rusagara, C. Harley / Applied Mathematics and Computation 238 (2014) 319–328 α=4
1 M=0.25 M=0.50 M=0.75
M=1.00
M=1.50
θ(x)
0.95
0.9
0.85 0
0.2
0.4
0.6
0.8
1
x
Fig. 3. Temperature distribution for an exponential fin profile where a ¼ 4 with n ¼ m ¼ 1=4 obtained via the relaxation scheme ð state solution ð Þ.
Þ compared to a steady
Fig. 4 depicts the fin efficiencies for both a rectangular fin and an exponential fin profile with parameter a ¼ 3. As depicted we find that the exponential fin profiles are more efficient than the rectangular fin profiles. This complies with our results which showed that the higher the exponential parameter the more efficient is the temperature distribution – see Figs. 1–3. The same results have been obtained from steady states solutions by [24] as shown in the graphics provided. Transient solutions are able to provide us with more than just solutions through time however. In fact, we believe that these solutions will be able to provide us with the transition time between two steady states which plays a major role in the evaluation of the fin performance. Thus far the fin performance is mostly described by the fin tip temperature rise, the fin efficiency and the base heat transfer rate [24] none of which consider the time taken by the temperature distribution process to reach some type of equilibrium. Even if it is difficult to know exactly how long a process may take to reach an equilibrium, the mean action time is able to provide us with an approximation of the total time taken as suggested by McNabb and Wake [21] and Landman and McGuiness [20]. Therefore, to better analyse the fin performance, we consider the mean action time as a means of obtaining an averaged transition time. In order to obtain smean , given how well our relaxation scheme performed against well known benchmark results, we use the same numerical scheme as a means of computing the mean action time as described in a previous section. This is done for small values of the thermo-geometric parameter, as per previous discussions, due to the nature of our equation. A computational measure, smax , of our time taken to equilibrium is obtained by fixing the maximum error between the iterants, i.e. we fix the error allowed as a means of defining computational convergence. This is required, given that in actuality, as per Landman and McGuiness [20], ‘‘diffusion processes take an infinite amount of time to come to equilibrium’’. We fix the tip temperature error allowed between the solutions as ¼ 5 103 . In order to compare smean and smax we employ the
Exponential Rectangular
1
η:Efficiency
0.8
0.6
0.4
0.2
0
1
2
3
4
5
M
6
7
8
9
10
Fig. 4. Comparison of the fin efficiency ðgÞ between a rectangular ða ¼ 0Þ and exponential ða ¼ 3Þ fin profile for n ¼ m ¼ 1=4 and varying M.
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I. Rusagara, C. Harley / Applied Mathematics and Computation 238 (2014) 319–328 Table 1
smean against M for a ¼ 0 and n ¼ m ¼ 1=4. M hs
mean
ð0Þ
hexact ð0Þ mean
%
s smax
0.25
0.50
0.75
1.00
1.50
66.94
66.91
66.80
66.61
65.88
0.6016 2.2784
0.5701 2.1306
0.5267 1.9149
0.4797 1.6677
0.3944 1.2039
Table 2
smean against M for a ¼ 1 and n ¼ m ¼ 1=4. M hs
mean
ð0Þ
hexact ð0Þ mean
%
s smax
0.25
0.50
0.75
1.00
1.50
66.95
66.92
66.86
66.77
66.44
0.3217 1.1856
0.3131 1.1461
0.3001 1.0847
0.28447 1.007
0.2507 0.8292
Table 3
smean against M for a ¼ 2 and n ¼ m ¼ 1=4. M hs
mean
ð0Þ
hexact ð0Þ
%
smean smax
0.25
0.50
0.75
1.00
1.50
66.90
66.88
66.82
66.76
66.52
0.1812 0.6492
0.1786 0.6353
0.1744 0.6133
0.1691 0.5847
0.1563 0.5155
temperature at the tip; this choice was provided by Turkyilmazoglou [24] where the tip temperature was classified among the most important characteristics of fins studied in engineering heat transfer problems. In many cases, our numerical results were taking extremely long to converge and as such the time taken to stabilize could not be determined with the hardware available to us. This again is an indication of the difficulties involved in finding the time taken for equilibrium to be reached. Our intention is to propose the mean action time as an additional performance index based on the duration of the process. Tables 1–3 show the mean action time for different types of exponential fin profiles. Table 1 is for a ¼ 0 (rectangular case), Table 2 for a ¼ 1 and Table 3 for a ¼ 2. For various values of the thermo-geometric parameter we give the ratio of the solution at the mean action time against the exact solution at the tip, the computed mean action time and the maximum time used for convergence of our numerical solution. We find that the highest times taken to approximately converge to the steady state analytical solutions were extremely high when compared to the mean action times that were computed. In fact we find that the tip temperature at the mean action time reaches approximately two thirds 23 of the tip temperature associated with the maximum time taken to convergence. Furthermore, we notice that smax =smean 3:7 (as M increases this ratio decreases slightly) across the various values of the thermo-geometric parameter which again shows that our methodology produces consistent results for small M. As such we find that, as per Landman and McGuiness’s suggestion [20], we have taken a more averaged approach and determined when the average temperature is a fixed fraction of the final equilibrium value, instead of obtaining the final time taken to reach the steady state. However the fact that this fraction is fixed at two thirds means that this information is useful in determining the time taken for the process to reach equilibrium and hence acts as a means of assessing the performance of the fin in terms of the length of time taken by this process. Furthermore, Tables 1–3 show that the mean action time taken to reach a steady state is less for the exponential profiles than for the rectangular case. It is seen that the higher the values of the exponential parameter a, the lower are the corresponding mean action times. This means that for applications that require the process to reach equilibrium fast it would be better to use exponential fin profiles with a larger exponential coefficient parameter. Therefore, one may select an appropriate profile which fulfills the requirement for reaching equilibrium at a certain speed; a useful performance indicator which can assist in this regard is the mean action time. 6. Conclusions The work was subdivided into two main parts: (1) establishing a relaxation system and numerically solving the corresponding system of equations and, (2) computing the mean action time to steady states and using this as an index for assessing the performance of the fin in terms of the time taken to reach some type of equilibrium. In our study, we have used a numerical relaxation scheme to solve a time dependent one dimensional heat transfer equation with different exponential fin profiles. The numerical relaxation scheme was implemented due to its simplicity and accuracy as described in the literature by Jin and Xin [15]. To the best of the authors’ knowledge this is the first time that this method has been employed within the context of such a problem. The results obtained complied with steady state
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analytical solutions [24] upon computational convergence. As such we were able to use this scheme to further investigate the mean action time of the process. Results obtained via the established numerical relaxation scheme showed that exponential fin profiles are more efficient in terms of heat transfer than the rectangular profile. The analysis of the mean action times obtained confirmed Landman and McGuiness’s suggestion that diffusive processes take a considerable amount of time to come to equilibrium [20]. We were able to show that the methodology we employed provided us with a fixed fraction of the final equilibrium temperature reached which becomes useful in assessing the performance of the fin profiles considered. Considering the mean action time along with the usually determined fin efficiency when evaluating the fin performance allows us to view the efficiency of a fin in a multi-dimensional fashion. 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