The efficiency, temperature distribution, and temperature at the tip of straight rectangular, growing and decaying moving exponential fins are investigated in this article. The influence of internal heat generation, surface and surrounding temperatures, convection-conduction, Peclet number and radiation-conduction is studied numerically on the efficiency, temperature profile, and temperature at the tip of the fin. Differential transform method is used to investigate the problem. It is revealed that thermal and thermo-geometric characteristics have a significant impact on the performance, temperature distribution, and temperature of the fin's tip.The results show that the temperature distribution of decaying exponential and rectangular fins is approximately 15 and 7% higher than growing exponential and rectangular fins respectively. It is estimated that the temperature distribution of the fin increases by approximately 6% when the porosity parameter is increased from 0.1 to 0.5. It is also observed that the decay exponential fin has better efficiency compared to growing exponential fin which offers significant advantages in mechanical engineering.
Citation: Zia Ud Din, Amir Ali, Zareen A. Khan, Gul Zaman. Heat transfer analysis: convective-radiative moving exponential porous fins with internal heat generation[J]. Mathematical Biosciences and Engineering, 2022, 19(11): 11491-11511. doi: 10.3934/mbe.2022535
[1] | Minlong Lin, Ke Tang . Selective further learning of hybrid ensemble for class imbalanced increment learning. Big Data and Information Analytics, 2017, 2(1): 1-21. doi: 10.3934/bdia.2017005 |
[2] | Subrata Dasgupta . Disentangling data, information and knowledge. Big Data and Information Analytics, 2016, 1(4): 377-390. doi: 10.3934/bdia.2016016 |
[3] | Qinglei Zhang, Wenying Feng . Detecting Coalition Attacks in Online Advertising: A hybrid data mining approach. Big Data and Information Analytics, 2016, 1(2): 227-245. doi: 10.3934/bdia.2016006 |
[4] | Tieliang Gong, Qian Zhao, Deyu Meng, Zongben Xu . Why Curriculum Learning & Self-paced Learning Work in Big/Noisy Data: A Theoretical Perspective. Big Data and Information Analytics, 2016, 1(1): 111-127. doi: 10.3934/bdia.2016.1.111 |
[5] | Xin Yun, Myung Hwan Chun . The impact of personalized recommendation on purchase intention under the background of big data. Big Data and Information Analytics, 2024, 8(0): 80-108. doi: 10.3934/bdia.2024005 |
[6] | Pankaj Sharma, David Baglee, Jaime Campos, Erkki Jantunen . Big data collection and analysis for manufacturing organisations. Big Data and Information Analytics, 2017, 2(2): 127-139. doi: 10.3934/bdia.2017002 |
[7] | Zhen Mei . Manifold Data Mining Helps Businesses Grow More Effectively. Big Data and Information Analytics, 2016, 1(2): 275-276. doi: 10.3934/bdia.2016009 |
[8] | Ricky Fok, Agnieszka Lasek, Jiye Li, Aijun An . Modeling daily guest count prediction. Big Data and Information Analytics, 2016, 1(4): 299-308. doi: 10.3934/bdia.2016012 |
[9] | M Supriya, AJ Deepa . Machine learning approach on healthcare big data: a review. Big Data and Information Analytics, 2020, 5(1): 58-75. doi: 10.3934/bdia.2020005 |
[10] | Sunmoo Yoon, Maria Patrao, Debbie Schauer, Jose Gutierrez . Prediction Models for Burden of Caregivers Applying Data Mining Techniques. Big Data and Information Analytics, 2017, 2(3): 209-217. doi: 10.3934/bdia.2017014 |
The efficiency, temperature distribution, and temperature at the tip of straight rectangular, growing and decaying moving exponential fins are investigated in this article. The influence of internal heat generation, surface and surrounding temperatures, convection-conduction, Peclet number and radiation-conduction is studied numerically on the efficiency, temperature profile, and temperature at the tip of the fin. Differential transform method is used to investigate the problem. It is revealed that thermal and thermo-geometric characteristics have a significant impact on the performance, temperature distribution, and temperature of the fin's tip.The results show that the temperature distribution of decaying exponential and rectangular fins is approximately 15 and 7% higher than growing exponential and rectangular fins respectively. It is estimated that the temperature distribution of the fin increases by approximately 6% when the porosity parameter is increased from 0.1 to 0.5. It is also observed that the decay exponential fin has better efficiency compared to growing exponential fin which offers significant advantages in mechanical engineering.
For a continuous risk outcome
Given fixed effects
In this paper, we assume that the risk outcome
y=Φ(a0+a1x1+⋯+akxk+bs), | (1.1) |
where
Given random effect model (1.1), the expected value
We introduce a family of interval distributions based on variable transformations. Probability densities for these distributions are provided (Proposition 2.1). Parameters of model (1.1) can then be estimated by maximum likelihood approaches assuming an interval distribution. In some cases, these parameters get an analytical solution without the needs for a model fitting (Proposition 4.1). We call a model with a random effect, where parameters are estimated by maximum likelihood assuming an interval distribution, an interval distribution model.
In its simplest form, the interval distribution model
The paper is organized as follows: in section 2, we introduce a family of interval distributions. A measure for tail fatness is defined. In section 3, we show examples of interval distributions and investigate their tail behaviours. We propose in section 4 an algorithm for estimating the parameters in model (1.1).
Interval distributions introduced in this section are defined for a risk outcome over a finite open interval
Let
Let
Φ:D→(c0,c1) | (2.1) |
be a transformation with continuous and positive derivatives
Given a continuous random variable
y=Φ(a+bs), | (2.2) |
where we assume that the range of variable
Proposition 2.1. Given
g(y,a,b)=U1/(bU2) | (2.3) |
G(y,a,b)=F[Φ−1(y)−ab]. | (2.4) |
where
U1=f{[Φ−1(y)−a]/b},U2=ϕ[Φ−1(y)] | (2.5) |
Proof. A proof for the case when
G(y,a,b)=P[Φ(a+bs)≤y] |
=P{s≤[Φ−1(y)−a]/b} |
=F{[Φ−1(y)−a]/b}. |
By chain rule and the relationship
∂Φ−1(y)∂y=1ϕ[Φ−1(y)]. | (2.6) |
Taking the derivative of
∂G(y,a,b)∂y=f{[Φ−1(y)−a]/b}bϕ[Φ−1(y)]=U1bU2. |
One can explore into these interval distributions for their shapes, including skewness and modality. For stress testing purposes, we are more interested in tail risk behaviours for these distributions.
Recall that, for a variable X over (−
For a risk outcome over a finite interval
We say that an interval distribution has a fat right tail if the limit
Given
Recall that, for a Beta distribution with parameters
Next, because the derivative of
z=Φ−1(y) | (2.7) |
Then
Lemma 2.2. Given
(ⅰ)
(ⅱ) If
(ⅲ) If
Proof. The first statement follows from the relationship
[g(y,a,b)(y1−y)β]−1/β=[g(y,a,b)]−1/βy1−y=[g(Φ(z),a,b)]−1/βy1−Φ(z). | (2.8) |
By L’Hospital’s rule and taking the derivatives of the numerator and the denominator of (2.8) with respect to
For tail convexity, we say that the right tail of an interval distribution is convex if
Again, write
h(z,a,b)=log[g(Φ(z),a,b)], | (2.9) |
where
g(y,a,b)=exp[h(z,a,b)]. | (2.10) |
By (2.9), (2.10), using (2.6) and the relationship
g′y=[h′z(z)/ϕ(z)]exp[h(Φ−1(y),a,b)],g″yy=[h″zz(z)ϕ2(z)−h′z(z)ϕ′z(z)ϕ3(z)+h′z(z)h′z(z)ϕ2(z)]exp[h(Φ−1(y),a,b)]. | (2.11) |
The following lemma is useful for checking tail convexity, it follows from (2.11).
Lemma 2.3. Suppose
In this section, we focus on the case where
One can explore into a wide list of densities with different choices for
A.
B.
C.
D.D.
Densities for cases A, B, C, and D are given respectively in (3.3) (section 3.1), (A.1), (A.3), and (A5) (Appendix A). Tail behaviour study is summarized in Propositions 3.3, 3.5, and Remark 3.6. Sketches of density plots are provided in Appendix B for distributions A, B, and C.
Using the notations of section 2, we have
By (2.5), we have
log(U1U2)=−z2+2az−a2+b2z22b2 | (3.1) |
=−(1−b2)(z−a1−b2)2+b21−b2a22b2. | (3.2) |
Therefore, we have
g(y,a,b)=1bexp{−(1−b2)(z−a1−b2)2+b21−b2a22b2}. | (3.3) |
Again, using the notations of section 2, we have
g(y,p,ρ)=√1−ρρexp{−12ρ[√1−ρΦ−1(y)−Φ−1(p)]2+12[Φ−1(y)]2}, | (3.4) |
where
Proposition 3.1. Density (3.3) is equivalent to (3.4) under the relationships:
a=Φ−1(p)√1−ρ and b=√ρ1−ρ. | (3.5) |
Proof. A similar proof can be found in [19]. By (3.4), we have
g(y,p,ρ)=√1−ρρexp{−1−ρ2ρ[Φ−1(y)−Φ−1(p)/√1−ρ]2+12[Φ−1(y)]2} |
=1bexp{−12[Φ−1(y)−ab]2}exp{12[Φ−1(y)]2} |
=U1/(bU2)=g(y,a,b). |
The following relationships are implied by (3.5):
ρ=b21+b2, | (3.6) |
a=Φ−1(p)√1+b2. | (3.7) |
Remark 3.2. The mode of
√1−ρ1−2ρΦ−1(p)=√1+b21−b2Φ−1(p)=a1−b2. |
This means
Proposition 3.3. The following statements hold for
(ⅰ)
(ⅱ)
(ⅲ) If
Proof. For statement (ⅰ), we have
Consider statement (ⅱ). First by (3.3), if
[g(Φ(z),a,b)]−1/β=b1/βexp(−(b2−1)z2+2az−a22βb2) | (3.8) |
By taking the derivative of (3.8) with respect to
−{∂[g(Φ(z),a,b)]−1β/∂z}/ϕ(z)=√2πb1β(b2−1)z+aβb2exp(−(b2−1)z2+2az−a22βb2+z22). | (3.9) |
Thus
{∂[g(Φ(z),a,b)]−1β/∂z}/ϕ(z)=−√2πb1β(b2−1)z+aβb2exp(−(b2−1)z2+2az−a22βb2+z22). | (3.10) |
Thus
For statement (ⅲ), we use Lemma 2.3. By (2.9) and using (3.2), we have
h(z,a,b)=log(U1bU2)=−(1−b2)(z−a1−b2)2+b21−b2a22b2−log(b). |
When
Remark 3.4. Assume
limz⤍+∞−{∂[g(Φ(z),a,b)]−1β/∂z}/ϕ(z) |
is
For these distributions, we again focus on their tail behaviours. A proof for the next proposition can be found in Appendix A.
Proposition 3.5. The following statements hold:
(a) Density
(b) The tailed index of
Remark 3.6. Among distributions A, B, C, and Beta distribution, distribution B gets the highest tailed index of 1, independent of the choices of
In this section, we assume that
First, we consider a simple case, where risk outcome
y=Φ(v+bs), | (4.1) |
where
Given a sample
LL=∑ni=1{logf(zi−vib)−logϕ(zi)−logb}, | (4.2) |
where
Recall that the least squares estimators of
SS=∑ni=1(zi−vi)2 | (4.3) |
has a closed form solution given by the transpose of
X=⌈1x11…xk11x12…xk2…1x1n…xkn⌉,Z=⌈z1z2…zn⌉. |
The next proposition shows there exists an analytical solution for the parameters of model (4.1).
Proposition 4.1. Given a sample
Proof. Dropping off the constant term from (4.2) and noting
LL=−12b2∑ni=1(zi−vi)2−nlogb, | (4.4) |
Hence the maximum likelihood estimates
Next, we consider the general case of model (1.1), where the risk outcome
y=Φ[v+ws], | (4.5) |
where parameter
(a)
(b)
Given a sample
LL=∑ni=1−12[(zi−vi)2/w2i−ui], | (4.6) |
LL=∑ni=1{−(zi−vi)/wi−2log[1+exp[−(zi−vi)/wi]−ui}, | (4.7) |
Recall that a function is log-concave if its logarithm is concave. If a function is concave, a local maximum is a global maximum, and the function is unimodal. This property is useful for searching maximum likelihood estimates.
Proposition 4.2. The functions (4.6) and (4.7) are concave as a function of
Proof. It is well-known that, if
For (4.7), the linear part
In general, parameters
Algorithm 4.3. Follow the steps below to estimate parameters of model (4.5):
(a) Given
(b) Given
(c) Iterate (a) and (b) until a convergence is reached.
With the interval distributions introduced in this paper, models with a random effect can be fitted for a continuous risk outcome by maximum likelihood approaches assuming an interval distribution. These models provide an alternative regression tool to the Beta regression model and fraction response model, and a tool for tail risk assessment as well.
Authors are very grateful to the third reviewer for many constructive comments. The first author is grateful to Biao Wu for many valuable conversations. Thanks also go to Clovis Sukam for his critical reading for the manuscript.
We would like to thank you for following the instructions above very closely in advance. It will definitely save us lot of time and expedite the process of your paper's publication.
The views expressed in this article are not necessarily those of Royal Bank of Canada and Scotiabank or any of their affiliates. Please direct any comments to Bill Huajian Yang at h_y02@yahoo.ca.
[1] |
M. Hatami, D. D. Ganji, M. Gorji-Bandpy, Experimental and numerical analysis of the optimized finned-tube heat exchanger for OM314 diesel exhaust exergy recovery, Energy Convers. Manage., 97 (2014), 26–41. https://doi.org/10.1016/j.enconman.2015.03.032 doi: 10.1016/j.enconman.2015.03.032
![]() |
[2] |
M. Ghazikhani, M. Hatami, B. Safari, The effect of alcoholic fuel additives on exergy parameters and emissions in a two-stroke gasoline engine, Arab. J. Sci. Eng., 39 (2014), 2117–2125. https://doi.org/10.1007/s13369-013-0738-3 doi: 10.1007/s13369-013-0738-3
![]() |
[3] |
M. Hatami, D. D. Ganji, Thermal performance of circular convective-radiative porous fins with different section shapes and materials, Energy Convers. Manage., 76 (2013), 185–193. https://doi.org/10.1016/j.enconman.2013.07.040 doi: 10.1016/j.enconman.2013.07.040
![]() |
[4] |
M. Turkyilmazoglu, Heat transfer from moving exponential fins exposed to heat generation, Int. J. Heat Mass Transfer, 116 (2018), 346–351. https://doi.org/10.1016/j.ijheatmasstransfer.2017.08.091 doi: 10.1016/j.ijheatmasstransfer.2017.08.091
![]() |
[5] |
E. Cuce, P. M. Cuce, Homotopy perturbation method for temperature distribution, fin efficiency and fin effectiveness of convective straight fins with temperature-dependent thermal conductivity, J. Mech. Eng. Sci., 227 (2013), 1754–1760. https://doi.org/10.1177/0954406212469579 doi: 10.1177/0954406212469579
![]() |
[6] | A. Y. Cengel, Introduction to Thermodynamics and Heat Transfer, Second Edition, McGraw-Hill Companies, 2008. |
[7] |
S. A. Atouei, K. Hosseinzadeh, M. Hatamic, S. E. Ghasemid, S. A. R. Sahebi, D. D. Ganji, Heat transfer study on convective-radiative semi-spherical fins with temperature-dependent properties and heat generation using efficient computational methods, Appl. Therm. Eng., 89 (2015), 299–305. https://doi.org/10.1016/j.applthermaleng.2015.05.084 doi: 10.1016/j.applthermaleng.2015.05.084
![]() |
[8] |
K. Hosseinzadeh, E. Montazer, M. B. Shafii, A. R. D. Ganji, Solidification enhancement in triplex thermal energy storage system via triplets fins configuration and hybrid nanoparticles, J. Energy Storage, 34 (2021), 102177. https://doi.org/10.1016/j.est.2020.102177 doi: 10.1016/j.est.2020.102177
![]() |
[9] | M. Hatami, D. D. Ganji, Optimization of the longitudinal fins with different geometries for increasing the heat transfer, in ISER 10th International Conference, Kuala Lumpur, Malaysia, 2015. |
[10] |
M. Turkyilmazoglu, Heat transfer from moving exponential fins exposed to heat generation, Int. J. Heat Mass Transfer, 116 (2018), 346–351. https://doi.org/10.1016/j.ijheatmasstransfer.2017.08.091 doi: 10.1016/j.ijheatmasstransfer.2017.08.091
![]() |
[11] |
B. Kundu, D. Bhanja, K. S. Lee, A model on the basis of analytics for computing maximum heat transfer in porous fins, Int. J. Heat Mass Transfer, 55 (2012), 7611–7622. https://doi.org/10.1016/j.ijheatmasstransfer.2012.07.069 doi: 10.1016/j.ijheatmasstransfer.2012.07.069
![]() |
[12] | Z. Din, A. Ali, S. Ullah, G. Zaman, Investigation of heat transfer from convective and radiative stretching/shrinking rectangular fins, Math. Probl. Eng., 2022 (2022). https://doi.org/10.1155/2022/1026698 |
[13] |
W. Ahmad, K. S. Syed, M. Ishaq, A. Hassan, Z. Iqbal, Numerical study of conjugate heat transfer in a double-pipe with exponential fins using DGFEM, Appl. Therm. Eng., 111 (2017), 1184–1201. https://doi.org/10.1016/j.applthermaleng.2016.09.171 doi: 10.1016/j.applthermaleng.2016.09.171
![]() |
[14] |
M. M. Rashidi, T. Hayat, T. Keimanesh, H. Yousefian, A study on heat transfer in a second-grade fluid through a porous medium with the modified differential transform method, Heat Transfer Asian Res., 42 (2013), 31–45. https://doi.org/10.1002/htj.21030 doi: 10.1002/htj.21030
![]() |
[15] | E. Erfani, M. M. Rashidi, A. B. Parsa. The modified differential transform method for solving off-centered stagnation flow toward a rotating disc, Int. J. Comput. Methods, 7 (2010), 655–670. https://doi.org/10.1142/S0219876210002404 |
[16] |
Y. Huang, X. Li, Exact and approximate solutions of convective-radiative fins with temperature-dependent thermal conductivity using integral equation method, Int. J. Heat Mass Transfer, 150 (2020), 119303. https://doi.org/10.1016/j.ijheatmasstransfer.2019.119303 doi: 10.1016/j.ijheatmasstransfer.2019.119303
![]() |
[17] | M. M. Rashidi, E. Erfani, New analytical method for solving Burgers and nonlinear heat transfer equations and comparison with HAM, Comput. Phys. Commun., 180 (2009), 1539–1544. |
[18] |
S. Panda, A. Bhowmik, R. Das, R. Repaka, S. C. Martha, Application of Homotopy analysis method and inverse solution of a rectangular wet fin, Energy Convers. Manage., 80 (2014), 303–318. https://doi.org/10.1016/j.cpc.2009.04.009 doi: 10.1016/j.cpc.2009.04.009
![]() |
[19] |
R. K. Singla, R. Das, Application of decomposition method and inverse prediction of parameters in a moving fin, Energy Convers. Manage., 84 (2014), 268–281. https://doi.org/10.1016/j.enconman.2014.04.045 doi: 10.1016/j.enconman.2014.04.045
![]() |
[20] |
C. Y. Zhang, X. F. Li, Temperature distribution of conductive-convective-radiative fins with temperature-dependent thermal conductivity, Int. Comm. Heat Mass Transfer, 117 (2020), 104799. https://doi.org/10.1016/j.icheatmasstransfer.2020.104799 doi: 10.1016/j.icheatmasstransfer.2020.104799
![]() |
[21] |
S. W. Sun, X. F. Li, Exact solution of the nonlinear fin problem with exponentially temperature-dependent thermal conductivity and heat transfer coefficient, Pramana J. Phys., 94 (2020), 1–10. https://doi.org/10.1007/s12043-020-01971-4 doi: 10.1007/s12043-020-01971-4
![]() |
[22] |
A. K. Asl, S. Hossainpour, M. M. Rashidi, M. A. Sheremet, Z. Yang, Comprehensive investigation of solid and porous fins influence on natural convection in an inclined rectangular enclosure, Int. J. Heat Mass Transfer, 133 (2019), 729–744. https://doi.org/10.1016/j.ijheatmasstransfer.2018.12.156 doi: 10.1016/j.ijheatmasstransfer.2018.12.156
![]() |
[23] |
S. Maalej, A. Zayoud, I. Abdelaziz, I. Saad, M. C. Zaghdoudi, Thermal performance of finned heat pipe system for Central Processing Unit cooling, Energy Convers. Manage., 218 (2020), 112977. https://doi.org/10.1016/j.enconman.2020.112977 doi: 10.1016/j.enconman.2020.112977
![]() |
[24] |
A. A. Joneidi, D. D. Ganji, M. Babaelahi, Differential Transformation Method to determine fin efficiency of convective straight fins with temperature dependent thermal conductivity, Int. Commun. Heat Mass Transfer, 36 (2009), 757–762. https://doi.org/10.1016/j.icheatmasstransfer.2009.03.020 doi: 10.1016/j.icheatmasstransfer.2009.03.020
![]() |
[25] |
C. H. Chiu, C. K. Chen, Applications of Adomian decomposition procedure to the analysis of convective radiative fins, J. Heat Transfer, 125 (2003), 312–316. https://doi.org/10.1115/1.1532012 doi: 10.1115/1.1532012
![]() |
[26] |
D. Lesnic, P. J. Heggs, A decomposition method for power-law fin-type problems, Int. Commun. Heat Mass Transfer, 31 (2004), 673–682. https://doi.org/10.1016/S0735-1933(04)00054-5 doi: 10.1016/S0735-1933(04)00054-5
![]() |
[27] |
R. Das, B. Kundu, Prediction of heat-generation and electromagnetic parameters from temperature response in porous fins, J. Thermophys. Heat Transfer, 35 (2021), 761–769. https://doi.org/10.2514/1.T6224 doi: 10.2514/1.T6224
![]() |
[28] |
R. Das, B. Kundu, Simultaneous estimation of heat generation and magnetic field in a radial porous fin from surface temperature information, Int. Commun. Heat Mass Transfer, 127 (2021), 105497. https://doi.org/10.1016/j.icheatmasstransfer.2021.105497 doi: 10.1016/j.icheatmasstransfer.2021.105497
![]() |
[29] |
D. Bhanja, B. Kundu, Thermal analysis of a constructal T-shaped porous fin with radiation effects, Int. J. Refrig., 31 (2011), 337–352. https://doi.org/10.1016/j.ijrefrig.2011.04.003 doi: 10.1016/j.ijrefrig.2011.04.003
![]() |
[30] |
B. Kundu, D. Bhanja, An analytical prediction for performance and optimum design analysis of porous fins, Int. J. Refrig., 31 (2011), 1483–1496. https://doi.org/10.1016/j.ijrefrig.2010.06.011 doi: 10.1016/j.ijrefrig.2010.06.011
![]() |
[31] |
R. Das, B. Kundu, Prediction of heat generation in a porous fin from surface temperature, J. Thermophys. Heat Transfer, 31 (2017), 781–790. https://doi.org/10.2514/1.T5098 doi: 10.2514/1.T5098
![]() |
[32] |
R. Das, Forward and inverse solutions of a conductive, convective and radiative cylindrical porous fin, Energy Convers. Manage., 87 (2014), 96–106. https://doi.org/10.1016/j.enconman.2014.06.096 doi: 10.1016/j.enconman.2014.06.096
![]() |
[33] | R. Das, D. K. Prasad. Prediction of porosity and thermal diffusivity in a porous fin using differential evolution algorithm, Swarm Evol. Comput., 23 (2015), 27–39. https://doi.org/10.1016/j.swevo.2015.03.001 |
[34] |
B. Kundu, S. J. Yook, An accurate approach for thermal analysis of porous longitudinal, spine and radial fins with all nonlinearity effects-analytical and unified assessment, Appl. Math. Comput., 402 (2021), 126124. https://doi.org/10.1016/j.amc.2021.126124 doi: 10.1016/j.amc.2021.126124
![]() |
[35] | G. A. Oguntala, R. A. Abd-Alhameed, G. M. Sobamowo, N. Eya, Effects of particles deposition on thermal performance of a convective-radiative heat sink porous fin of an electronic component, Therm. Sci. Eng. Prog., 6 (2018), 177–185. https://doi.org/10.1016/j.tsep.2017.10.019 |
[36] |
M. A. Vatanparast, S. Hossainpour, A. Keyhani-Asl, S. Forouzi, Numerical investigation of total entropy generation in a rectangular channel with staggered semi-porous fins, Int. Commun. Heat Mass Transfer, 111 (2020), 104446. https://doi.org/10.1016/j.icheatmasstransfer.2019.104446 doi: 10.1016/j.icheatmasstransfer.2019.104446
![]() |
[37] |
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. https://doi.org/10.1016/j.ijthermalsci.2011.12.019 doi: 10.1016/j.ijthermalsci.2011.12.019
![]() |
[38] |
Z. U. Din, A. Ali, G. Zaman, Entropy generation in moving exponential porous fins with natural convection, radiation and internal heat generation, Arch. Appl. Mech., 92 (2022), 933–944. https://doi.org/10.1007/s00419-021-02081-2 doi: 10.1007/s00419-021-02081-2
![]() |
[39] | Z. U. Din, A. Ali, M. D. la Sen, G. Zaman, Entropy generation from convective-radiative moving exponential porous fins with variable thermal conductivity and internal heat generations, Sci. Rep., 12 (2022), 1791. https://doi.org/10.1038/s41598-022-05507-1 |
[40] |
M. Hatami, A. Hasanpour, D. D. Ganji, Heat transfer study through porous fins (Si3N4 and AL) with temperature-dependent heat generation, Energy Convers. Manage., 74 (2013), 9–16. https://doi.org/10.1016/j.enconman.2013.04.034 doi: 10.1016/j.enconman.2013.04.034
![]() |
[41] |
M. Hatami, D. D. Ganji, Thermal and flow analysis of microchannel heat sink (MCHS) cooled by Cu-water nanofluid using porous media approach and least square method, Energy Convers. Manage., 78 (2014), 347–358. https://doi.org/10.1016/j.enconman.2013.10.063 doi: 10.1016/j.enconman.2013.10.063
![]() |
[42] |
M. Turkyilmazoglu, Thermal performance of optimum exponential fin profiles subjected to a temperature jump, Int. J. Numer. Methods Heat Fluid Flow, 32 (2021), 1002–1011. https://doi.org/10.1108/HFF-02-2021-0132 doi: 10.1108/HFF-02-2021-0132
![]() |
[43] | A. R. A. Khaled, Thermal characterizations of exponential fin systems, Math. Probl. Eng., (2010), 765729. https://doi.org/10.1155/2010/765729 |
[44] |
M. F. Najafabadi, H. T. Rostami, K. Hosseinzadeh, D. D. Ganji, Thermal analysis of a moving fin using the radial basis function approximation, Heat Transfer, 50 (2021), 7553–7567. https://doi.org/10.1002/htj.22242 doi: 10.1002/htj.22242
![]() |
[45] |
S. Hosseinzadeh, K. Hosseinzadeh, A. Hasibi, D. D. Ganji, Thermal analysis of moving porous fin wetted by hybrid nanofluid with trapezoidal, concave parabolic and convex cross sections, Case Stud. Therm. Eng., 30 (2022), 101757. https://doi.org/10.1016/j.csite.2022.101757 doi: 10.1016/j.csite.2022.101757
![]() |
[46] | M. A. E. Moghaddam, M. R. H. S. Abandani, K. Hosseinzadeh, M. B. Shafii, D. D. Ganji, Metal foam and fin implementation into a triple concentric tube heat exchanger over melting evolution, Theor. Appl. Mech. Lett., (2022), 100332. https://doi.org/10.1016/j.taml.2022.100332 |
[47] | B. Jalili, N. Aghaee, P. Jalili, D. D. Ganji, Novel usage of the curved rectangular fin on the heat transfer of a double-pipe heat exchanger with a nanofluid, Case Stud. Therm., (2022), 102086. https://doi.org/10.1016/j.csite.2022.102086 |
[48] |
B. Jalili, S. Sadighi, P. Jalili, D. D. Ganji, Characteristics of ferrofluid flow over a stretching sheet with suction and injection, Case Stud. Therm., 14 (2019), 100470. https://doi.org/10.1016/j.csite.2022.102086 doi: 10.1016/j.csite.2022.102086
![]() |
[49] |
B. Jalili, S. Sadighi, P. Jalili, D. D. Ganji, Effect of magnetic and boundary parameters on flow characteristics analysis of micropolar ferrofluid through the shrinking sheet with effective thermal conductivity, Chin. J. Phys., 71 (2021), 136–150. https://doi.org/10.1016/j.cjph.2020.02.034 doi: 10.1016/j.cjph.2020.02.034
![]() |
[50] |
P. Jalili, D. D. Ganji, B. Jalili, D. D. Ganji, Evaluation of electro-osmotic flow in a nanochannel via semi-analytical method, Therm. Sci., 16 (2012), 1297–1302. http://DOI:10.2298/TSCI1205297J doi: 10.2298/TSCI1205297J
![]() |
[51] |
M. Turkyilmazoglu, Efficiency of heat and mass transfer in fully wet porous fins: exponential fins versus straight fins, Int. J. Refrig., 46 (2014), 158–164. https://doi.org/10.1016/j.ijrefrig.2014.04.011 doi: 10.1016/j.ijrefrig.2014.04.011
![]() |
[52] |
B. Kundu, K. S. Lee, Analytic solution for heat transfer of wet fins on account of all nonlinearity effects, Energy, 41 (2012), 354–367. https://doi.org/10.1016/j.energy.2012.03.004 doi: 10.1016/j.energy.2012.03.004
![]() |
[53] |
R. das, K. T. Ooi, Predicting multiple combination of parameters for designing a porous fin subjected to a given temperature requirement, Energy Convers. Manage., 66 (2013), 211–219. https://doi.org/10.1016/j.enconman.2012.10.019 doi: 10.1016/j.enconman.2012.10.019
![]() |
[54] |
M. Torabi A. Aziz, K. Zhang, A comparative study of longitudinal fins of rectangular, trapezoidal and concave parabolic profiles with multiple nonlinearities, Energies, 51 (2013), 243–256. https://doi.org/10.1016/j.energy.2012.11.052 doi: 10.1016/j.energy.2012.11.052
![]() |