Modeling of Heat Distribution in Porous Aluminum Using Fractional Differential Equation
Abstract
:1. Introduction
2. Fractional Heat Conduction Equation
3. Formulation of the Problem
4. Method of Solution
4.1. Solution of the Direct Problem
4.2. Minimum of the Functional
$F\left(\mathbf{x}\right)$  minimized function, $\mathbf{x}=({x}_{1},\dots ,{x}_{n})\in D$ 
n  dimension (number of variables) 
$nT$  number of threads 
$M=nT\xb7p$  number of ants in population 
I  number of iterations 
L  number of pheromone spots 
$q,\xi $  parameters of the algorithm 
Algorithm 1: Parallel Real ACO algorithm  
Initialization of the algorithm  
1.  Setting input parameters of the algorithm L, M, I, $nT$, q, $\xi $. 
2.  Randomly generate L pheromone spots (solutions) and assign them to set ${T}_{0}$ (starting archive). 
3.  Calculate values of the minimized function F for each pheromone spot and sort the archive ${T}_{0}$ from best to worst solution. 
Iterative process  
4.  Assigning probabilities to pheromone spots (solutions) according to the following formula:
$${p}_{l}=\frac{{\omega}_{l}}{{\sum}_{l=1}^{L}{\omega}_{l}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}l=1,2,\dots ,$$
$${\omega}_{l}=\frac{1}{qL\sqrt{2\pi}}\xb7{e}^{\frac{{(l1)}^{2}}{2{q}^{2}{L}^{2}}}.$$

5.  Ant chooses a random lth solution with probability ${p}_{l}$. 
6.  Ant transforms the jth coordinate ($j=1,2,\dots ,n$) of lth solution ${s}_{j}^{l}$ by sampling proximity with the probability density function (Gaussian function)
$$g(x,\mu ,\sigma )=\frac{1}{\sigma \sqrt{2\pi}}\xb7{e}^{\frac{{(x\mu )}^{2}}{2{\sigma}^{2}}},$$

7.  Repeat steps 5–6 for each ant. We obtain M new solutions (pheromone spots). 
8.  Divide new solutions on $nT$ groups. Calculate values of minimized function F for each new solution (parallel computing). 
9.  Add to the archive ${T}_{i}$ new solutions, sort the archive by quality of solutions, remove M worst solution. 
10.  Repeat steps 4–9 I times. 
5. Results
 $\lambda ={a}_{1}\phantom{\rule{0.166667em}{0ex}}\left[\frac{J}{{s}^{\alpha}\xb7\mathrm{m}\xb7\mathrm{K}}\right]$—modified thermal conductivity coefficient,
 $f\left(x\right)={a}_{2}\phantom{\rule{0.166667em}{0ex}}\left[\mathrm{K}\right]$—initial condition,
 $h\left(t\right)={a}_{3}{t}^{2}+{a}_{4}t+{a}_{5}\phantom{\rule{0.166667em}{0ex}}\left[\frac{W}{{\mathrm{m}}^{2}\xb7\mathrm{K}}\right]$—heat transfer coefficient,
 $\alpha ={a}_{6}$—order of derivative,
6. Conclusions
Acknowledgments
Author Contributions
Conflicts of Interest
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100 × 1995  ${\mathbf{\sigma}}_{{\mathit{a}}_{\mathit{i}}}$  100 × 3990  ${\mathbf{\sigma}}_{{\mathit{a}}_{\mathit{i}}}$  

${a}_{1}$  300.00  69.74  237.91  67.78 
${a}_{2}$  569.73  2.02  566.74  3.80 
${a}_{3}$  1.63  0.40  1.52  2.20 
${a}_{4}$  4.72  0.67  5.00  4.27 
${a}_{5}$  198.02  46.05  178.05  51.73 
${a}_{6}$  0.20  0.05  0.21  0.09 
value of the functional  246.98  352.88 
100 × 1995  100 × 3990  

${\Delta}_{\mathrm{avg}}\left[K\right]$  4.92  4.77 
${\Delta}_{\mathrm{max}}\left[K\right]$  11.04  12.38 
${\delta}_{\mathrm{avg}}[\%]$  1.06  1.02 
${\delta}_{\mathrm{max}}[\%]$  3.08  3.46 
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Brociek, R.; Słota, D.; Król, M.; Matula, G.; Kwaśny, W. Modeling of Heat Distribution in Porous Aluminum Using Fractional Differential Equation. Fractal Fract. 2017, 1, 17. https://doi.org/10.3390/fractalfract1010017
Brociek R, Słota D, Król M, Matula G, Kwaśny W. Modeling of Heat Distribution in Porous Aluminum Using Fractional Differential Equation. Fractal and Fractional. 2017; 1(1):17. https://doi.org/10.3390/fractalfract1010017
Chicago/Turabian StyleBrociek, Rafał, Damian Słota, Mariusz Król, Grzegorz Matula, and Waldemar Kwaśny. 2017. "Modeling of Heat Distribution in Porous Aluminum Using Fractional Differential Equation" Fractal and Fractional 1, no. 1: 17. https://doi.org/10.3390/fractalfract1010017