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    Journals >High Power Laser and Particle Beams >Volume 36 >Issue 1 >Page 013014 > Article
    • High Power Laser and Particle Beams
    • Vol. 36, Issue 1, 013014 (2024)
    Simulation and verification of 3D temperature model for high power microwave heating
    Hao Wu, Shaofu Li*, Wei Wang, Cheng Jiang, and Yingying Tang
    Author Affiliations
  • School of Information Engineering, Southwest University of Science and Technology, Mianyang 621010, China
    • show less
    DOI: 10.11884/HPLPB202436.230281 Cite this Article
    Hao Wu, Shaofu Li, Wei Wang, Cheng Jiang, Yingying Tang. Simulation and verification of 3D temperature model for high power microwave heating[J]. High Power Laser and Particle Beams, 2024, 36(1): 013014 Copy Citation Text show less

    Abstract

    Microwave heating inhomogeneity has always been a hot issue in microwave heating control research. According to the physical structure of the microwave heating device, the static difference model of the surface temperature of each layer in the furnace is established, and the actual power of microwave heating is obtained by combining with experiments. Based on the finite difference method of heat transfer, the temperature distribution model in 3D space is established, and the effectiveness of the model is verified by MATLAB and COMSOL simulations. The equilibrium temperature of the heated medium obtained by uniformly heating the microwave is compared with the temperature distribution during uneven heating, the partial temperature rise equilibrium points of the medium during the microwave heating process are identified. Finally, comparison is carried out to find out the best point for the control object for expert PID (proportion-integral-derivative) microwave heating. The experimental results show that this method can accurately measure the equilibrium temperature of liquid heated medium at any time, and can make microwave heating more widely used in industrial production.

    Keywords

    $ {Q_{{\mathrm{San}}\_k}} = {\left( {\dfrac{1}{{\left( {{h_1} + {h_2}} \right)}} + \dfrac{\delta }{\lambda } + \dfrac{1}{{{h_3}}}} \right)^{ - 1}} \cdot \left( {{T_k} - {T_{k - 1}}} \right) \cdot A{\text{ = }}K \cdot \left( {{T_k} - {T_{k - 1}}} \right) \cdot A $(1)

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    $ {T_k} = {T_{k - 1}} + \dfrac{{{Q_k} - {Q_{{\mathrm{San}}\_k - 1}}}}{{{{{c}}_1}{m_1} + {{{c}}_2}{m_2}}} $(2)

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    $ {k_{{\mathrm{delay}}}} = 10{m_1} + 50 $(3)

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    $ P\left( k \right) = \left\{ \begin{gathered} {Q_k},\quad {T_k} \lt {T_{sk}} \\ 0,\;\;\quad {T_k} \geqslant {T_{sk}} \\ \end{gathered} \right. $(4)

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    $ \rho c\dfrac{\partial T}{\partial t}=\lambda\Bigg(\dfrac{1}{r}\dfrac{\partial T}{\partial r}+\dfrac{\partial^2T}{\partial r^2}+\dfrac{1}{r^2}\dfrac{\partial^2T}{\partial\varphi^2}+\dfrac{\partial^2T}{\partial\text{z}^2}\Bigg)+\mathit{\Phi} $(5)

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    $ \rho c\dfrac{\partial T}{\partial t}=\lambda\Bigg(\dfrac{1}{r}\dfrac{\partial T}{\partial r}+\dfrac{\partial^2T}{\partial r^2}+\dfrac{\partial^2T}{\partial\text{z}^2}\Bigg)+\mathit{\Phi} $(6)

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    $ \begin{split} & \rho_1c_1\dfrac{T\left(j,k,n+1\right)-T\left(j,k,n\right)}{\Delta\tau}=\lambda_1\Bigg(\dfrac{1}{r_j}\dfrac{T\left(j+1,k,n\right)-T\left(j,k,n\right)}{\Delta r}+ \\ & \quad\quad\dfrac{T\left(j+1,k,n\right)-2T\left(j,k,n\right)+T\left(j-1,k,n\right)}{\left(\Delta r\right)^2}+\dfrac{T\left(j,k+1,n\right)-2T\left(j,k,n\right)+T\left(j,k-1,n\right)}{\left(\Delta\text{z}\right)^2}\Bigg)+\mathit{\Phi}\end{split} $(7)

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    $ \begin{split}T\left(j,k,n+1\right)= & \left(\dfrac{1}{j}+1\right)\cdot f_1\cdot T\left(j+1,k,n\right)-\left(\dfrac{1}{j}\cdot f_1+2f_1+2g_1-1\right)\cdot T\left(j,k,n\right)+ \\ & f_1\cdot T\left(j-1,k,n\right)+g_1\cdot\left(T\left(j,k-1,n\right)+T\left(j,k+1,n\right)\right)+\dfrac{e_1\mathit{\Phi}}{\lambda_1}\end{split} $(8)

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    $ \begin{split} & \dfrac{{\partial T\left( {1,k,n + 1} \right)}}{{\partial r}} = 0;\quad r = \Delta r \\ & {\lambda _1}\dfrac{{\partial T\left( {j,k,n + 1} \right)}}{{\partial r}} = {\lambda _2}\dfrac{{\partial T\left( {j + 1,k,n + 1} \right)}}{{\partial r}};\quad r = {r_1} \end{split} $(9)

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    $ \begin{split} &T\left( {1,k,n + 1} \right) = T\left( {2,k,n + 1} \right);\quad r = \Delta r \\ & T\left( {j + 1,k,n + 1} \right) = \dfrac{{{p_2}}}{{\left( {{p_1} + {p_2}} \right)}} \cdot T\left( {j + 2,k,n + 1} \right) + \dfrac{{{p_1}}}{{\left( {{p_1} + {p_2}} \right)}} \cdot T\left( {j,k,n + 1} \right);\quad r = {r_1} \end{split}$(10)

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    $ - {\lambda _4}\dfrac{{\partial T\left( {j,k,n + 1} \right)}}{{\partial r}} = h\left( {T\left( {j + 1,k,n + 1} \right) - {T_{{\mathrm{s}}\_{\mathrm{right}}}}} \right)；r = {r_4} $(11)

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    $ T\left( {j + 1,k,n + 1} \right) = \dfrac{d}{{1 + d}} \cdot T\left( {j,k,n + 1} \right) + \dfrac{{{T_{{\mathrm{s}}\_{\mathrm{right}}}}}}{{1 + d}} $(12)

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    $ F{o_\Delta } = \dfrac{{\lambda \cdot \Delta \tau }}{{\rho \cdot c \cdot {{\left( {\Delta r} \right)}^2}}} \leqslant 0.5；F{o_\Delta } \leqslant \dfrac{1}{{2\left( {1 + B{i_\Delta }} \right)}}；B{i_\Delta } = \dfrac{{h \cdot \Delta r}}{{{\lambda _4}}} $(13)

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    $ \mathit{\Phi}=\dfrac{Q_k}{V\mathrm{_{\mathit{\mathrm{water}}}}}=\dfrac{3\; 830}{3.14\times\left(0.08\right)^2\times0.24}\approx794\; 105\left(\mathrm{W}/\mathrm{m}^3\right) $(14)

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    Hao Wu, Shaofu Li, Wei Wang, Cheng Jiang, Yingying Tang. Simulation and verification of 3D temperature model for high power microwave heating[J]. High Power Laser and Particle Beams, 2024, 36(1): 013014
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