A Novel Approach for Simulation and Optimization of Rubber Vulcanization
Abstract
:1. Introduction
2. Materials and Methods
2.1. Materials
2.2. Simulation
 Minimal vulcanization degree (${\alpha}_{min}$);
 Average vulcanization degree (${\alpha}_{a}$);
 Maximal vulcanization degree (${\alpha}_{max}$);
 Minimal temperature (${T}_{min}$);
 Average temperature (${T}_{a}$);
 Maximal temperature (${T}_{max}$);
 Minimal vulcanization rate (${v}_{min}$);
 Average vulcanization rate (${v}_{a}$);
 Maximal vulcanization rate (${v}_{max}$).
3. Results and Discussion
3.1. Simulation of Vulcanization of a Sphere of Different Dimensions
3.2. Simulation of Vulcanization of Rubber Wheels
4. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
List of Variables and Their Definitions
References
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Heat Transfer  

Heat transfer equation 
$$\rho {C}_{p}\frac{\partial T}{\partial t}+\nabla \mathit{q}=Q$$

Conductivity 
$$\mathit{q}=\lambda \nabla T$$

$$\nabla \mathit{q}=\frac{1}{r}\frac{\partial}{\partial r}\left(r\lambda \frac{\partial T}{\partial r}\right)+\frac{\partial}{\partial z}\left(\lambda \frac{\partial T}{\partial z}\right)$$
 
Convection 
$$\mathit{n}\xb7\mathit{q}=h({T}_{amb}T)$$

Heat Source 
$$Q=\frac{d{\alpha}_{c}}{dt}{q}_{v}\rho $$

Limit conditions and values 
$$h=\left\{\begin{array}{c}10,000\phantom{\rule{0.166667em}{0ex}}{\mathrm{W}\mathrm{m}}^{2}\phantom{\rule{0.166667em}{0ex}}{\mathrm{K}}^{1},{\alpha}_{a}<0.9\hfill \\ 11.4\phantom{\rule{0.166667em}{0ex}}{\mathrm{W}\mathrm{m}}^{2}\phantom{\rule{0.166667em}{0ex}}{\mathrm{K}}^{1},{\alpha}_{a}\ge 0.9\hfill \end{array}\right.$$

$${T}_{amb}=\left\{\begin{array}{c}{T}_{M},{\alpha}_{a}<0.9\hfill \\ {T}_{0},{\alpha}_{a}\ge 0.9\hfill \end{array}\right.$$
 
${T}_{0}$ = 25 °C  
${T}_{M}$  
${C}_{p}$ = 1.82 kJ kg${}^{1}$ K${}^{1}$ [38]  
$\rho $ = 1020 kg m${}^{3}$ [38]  
$\lambda $ = 0.28 W m${}^{1}$ K${}^{1}$ [38]  
K = 0.15 mm${}^{2}$ s${}^{1}$ [38]  
${q}_{v}$ = 13 kJ kg${}^{1}$ [40]  
Solutions  T(r, z, t) 
${T}_{min}\left(t\right)$, ${T}_{a}\left(t\right)$, ${T}_{max}\left(t\right)$  
Kinetic model  
Equations 
$$\frac{d{\alpha}_{c}(r,z,t)}{dt}={A}_{c}{e}^{\frac{{E}_{ac}}{RT(r,z,t)}}{\alpha}_{c}{(r,z,t)}^{\frac{n\left(T\right)1}{n\left(T\right)}}{(1{\alpha}_{c}(r,z,t))}^{\frac{n\left(T\right)+1}{n\left(T\right)}}$$

$$\frac{d{\alpha}_{r}(r,z,t)}{dt}={A}_{r}{e}^{\frac{{E}_{ar}}{RT(r,z,t)}}(x\left(T\right){\alpha}_{r}(r,z,t))$$
 
$$\alpha (r,z,t)={\alpha}_{c}(r,z,t){\alpha}_{r}(r,z,t)$$
 
Limit conditions and values 
$$n(r,z,t)=\left\{\begin{array}{c}1,{K}_{n}T(r,z,t)+{N}_{n}\le 1\hfill \\ {K}_{n}T(r,z,t)+{N}_{n},{K}_{n}T(r,z,t)+{N}_{n}>1\hfill \end{array}\right.$$

$$x(r,z,t)=\left\{\begin{array}{c}0,{K}_{x}T(r,z,t)+{N}_{x}\le 0\hfill \\ {K}_{x}T(r,z,t)+{N}_{x},{K}_{x}T(r,z,t)+{N}_{x}>0\hfill \end{array}\right.$$
 
${N}_{n}$ = −5.335 [38]  
${K}_{n}$ = 0.019662 K${}^{1}$ [38]  
${N}_{x}$ = −1.479744 [38]  
${K}_{x}$ = 0.003896 K${}^{1}$ [38]  
${E}_{ac}$ = 85.008 kJ mol${}^{1}$ [38]  
${A}_{c}$ = 1.017 × 10${}^{8}$ [38]  
${E}_{ar}$ = 69.187 kJ mol${}^{1}$ [38]  
${A}_{r}$ = 8.639 × 10${}^{4}$ [38]  
Solutions  $\alpha (r,z,t)$ 
${\alpha}_{min}\left(t\right)$, ${\alpha}_{sr}\left(t\right)$, ${\alpha}_{max}\left(t\right)$  
${v}_{min}\left(t\right)$, ${v}_{sr}\left(t\right)$, ${v}_{max}\left(t\right)$ 
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Lubura, J.; Kojić, P.; Pavličević, J.; Ikonić, B.; Balaban, D.; Bera, O. A Novel Approach for Simulation and Optimization of Rubber Vulcanization. Polymers 2023, 15, 1750. https://doi.org/10.3390/polym15071750
Lubura J, Kojić P, Pavličević J, Ikonić B, Balaban D, Bera O. A Novel Approach for Simulation and Optimization of Rubber Vulcanization. Polymers. 2023; 15(7):1750. https://doi.org/10.3390/polym15071750
Chicago/Turabian StyleLubura, Jelena, Predrag Kojić, Jelena Pavličević, Bojana Ikonić, Dario Balaban, and Oskar Bera. 2023. "A Novel Approach for Simulation and Optimization of Rubber Vulcanization" Polymers 15, no. 7: 1750. https://doi.org/10.3390/polym15071750