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Nostatsionar issiqlik o‘tkazuvchanligi tahlili

Chekli elementlar usuli (Finite Element Method) yordamida qattiq jismdagi issiqlik o‘tkazuvchanligini tahlil qilish uchun vaqt diskretlashtirishi va iterativ yechish usuli ko‘rsatiladi. Kontinuumning boshqaruvchi tenglamalari va chegaraviy shartlari uchun Issiqlik o‘tkazuvchanligi tenglamasi ga qarang.

Diskretlashtirilgan tenglama (boshlang‘ich nuqta)

Issiqlik o‘tkazuvchanligi tenglamasini (issiqlik o‘tkazuvchanligi tenglamasi (gov_he_main)) Galerkin usuli bilan diskretlashtirsak,

\[\begin{equation} K T + M \frac{\partial T}{\partial t} = F \label{eq:2.4.8} \end{equation}\]

bu yerda,

\[\begin{equation} K = \int\left( k_x \frac{\partial N^T}{\partial x}\frac{\partial N}{\partial x} + k_y \frac{\partial N^T}{\partial y}\frac{\partial N}{\partial y} + k_z \frac{\partial N^T}{\partial z}\frac{\partial N}{\partial z} \right) dV + \int hc N^T N ds + \int hr N^T N ds \label{eq:2.4.9} \end{equation}\]
\[\begin{equation} M = \int \rho c N^T N dV \label{eq:2.4.10} \end{equation}\]
\[\begin{equation} F = \int Q N^T dV - \int q_s N^T dS + \int{hc} T c N^T dS + \int{hcTr} ({T+Tr}) ({T^2 + T r^2}) N^T dS \label{eq:2.4.11} \end{equation}\]
\[\begin{equation} N = (N^1, N^2, \ldots, Ni) \label{eq:2.4.12} \end{equation}\]

Bu yerda \(K\), \(M\), \(F\), \(N\) mos ravishda issiqlik o‘tkazuvchanligi matritsasi (chegaraviy hissalardagi konveksiya va nurlanish hadlarini ham o‘z ichiga oladi), massa matritsasi, issiqlik yuk vektori va shakl funksiyalari matritsasidir. Material xossalari belgilarining (\(\rho\), \(c\), \(k_x, k_y, k_z\), \(Q\), \(hc\), \(hr\) va boshqalar) ta’riflari Issiqlik o‘tkazuvchanligi tenglamasi ga muvofiq.

Vaqt diskretlashtirishi va iterativ yechish usuli

\(\eqref{eq:2.4.8}\) tenglama nochiziqli va nostatsionardir. Vaqt bo‘yicha Backward Euler usuli bilan diskretlashtirib, \(t=t_0\) vaqtdagi harorat ma’lum bo‘lganda \(t=t_0+\Delta t\) vaqtdagi haroratni quyidagi tenglama yordamida hisoblaymiz.

\[\begin{equation} K_{t=t_0+\Delta t} T_{t=t_0+\Delta t} + M_{t=t_0+\Delta t} \frac{T_{t=t_0+\Delta t} - T_{t=t_0}}{\Delta t} = F_{t=t_0+\Delta t} \label{eq:2.4.13} \end{equation}\]

Bu yerda \(\eqref{eq:2.4.13}\) tenglamani taxminan qanoatlantiruvchi harorat vektori \(T_{t=t_0+\Delta t}^{(i)}\) ni yaxshilab, aniqroq yechim \(T_{t=t_0+\Delta t}^{(i)+1}\) ni topamiz.

Buning uchun avval harorat vektorini quyidagicha ifodalaymiz.

\[\begin{equation} T_{t=t_0+\Delta t}= T_{t=t_0+\Delta t}^{(i)} + \Delta T_{t=t_0+\Delta t}^{(i)} \label{eq:2.4.14} \end{equation}\]

Issiqlik o‘tkazuvchanligi matritsasi bilan harorat vektori ko‘paytmasi, massa matritsasi va boshqalarni quyidagicha taxminiy ifodalaymiz.

\[\begin{equation} K_{t=t_0+\Delta t} T_{t=t_0+\Delta t} = K_{t=t_0+\Delta t}^{(i)} T_{t=t_0+\Delta t}^{(i)} + \frac{\partial \big(K_{t=t_0+\Delta t}^{(i)} T_{t=t_0+\Delta t}^{(i)}\big) } {\partial T_{t=t_0+\Delta t}^{(i)} } \Delta T_{t=t_0+\Delta t}^{(i)} \label{eq:2.4.15} \end{equation}\]
\[\begin{equation} M_{t=t_0+\Delta t} = M_{t=t_0+\Delta t}^{(i)} + \frac{\partial M_{t=t_0+\Delta t}^{(i)}}{\partial T_{t=t_0+\Delta t}^{(i)}} \Delta T_{t=t_0+\Delta t}^{(i)} \label{eq:2.4.16} \end{equation}\]

\(\eqref{eq:2.4.14}\), \(\eqref{eq:2.4.15}\) va \(\eqref{eq:2.4.16}\) tenglamalarni \(\eqref{eq:2.4.13}\) ga qo‘yib, ikkinchi va undan yuqori tartibli hadlarni tashlab yuborsak, quyidagi tenglamani olamiz.

\[\begin{equation} \bigg(\frac{M_{t=t_0+\Delta t}^{(i)}}{\Delta t} + \frac {\partial M_{t=t_0+\Delta t}^{(i)} } { \partial T_{t=t_0+\Delta t}^{(i)} } \frac{T_{t=t_0+\Delta t}^{(i)} - T_{t=t_0}}{\Delta t} + \frac{\partial \big(K_{t=t_0+\Delta t}^{(i)} T_{t=t_0+\Delta t}^{(i)}\big)} {\partial T_{t=t_0+\Delta t}^{(i)}} \bigg) \Delta T_{t=t_0+\Delta t}^{(i)} \\\ = F_{t=t_0+\Delta t} - M_{t=t_0+\Delta t}^{(i)} \frac{T_{t=t_0+\Delta t}^{(i)} - T_{t=t_0}}{\Delta t} - K_{t=t_0+\Delta t}^{(i)} T_{t=t_0+\Delta t}^{(i)} \label{eq:2.4.17} \end{equation}\]

Bundan tashqari, chap tomondagi koeffitsiyent matritsasini quyidagi tenglama orqali taxminiy baholaymiz.

\[\begin{equation} K^{(i)} = \frac{M_{t=t_0+\Delta t}^{(i)}}{\Delta t} + \frac{\partial \big( K_{t=t_0+\Delta t}^{(i)} T_{t=t_0+\Delta t}^{(i)} \big)}{\partial T^{(i)}_{t=t_0+\Delta t}} = \frac{M_{t=t_0+\Delta t}^{(i)}}{\Delta t} + K_{T_{t=t_0+\Delta t}}^{(i)} \label{eq:2.4.18} \end{equation}\]

Bu yerda \(K_{T_{t=t_0+\Delta t}}^{(i)}\) tangensial qattiqlik matritsasidir.

Natijada quyidagi tenglama bo‘yicha iterativ hisoblash orqali \(t=t_0+\Delta t\) vaqtdagi haroratni hisoblash mumkin.

\[\begin{equation} K^{(i)} \Delta T_{t=t_0+\Delta t}^{(i)} = F_{t=t_0+\Delta t} - M_{t=t_0+\Delta t}^{(i)} \frac{T_{t=t_0+\Delta t}^{(i)} - T_{t=t_0}}{\Delta t} - K^{(i)} T_{t=t_0+\Delta t}^{(i)} \label{eq:2.4.19} \end{equation}\]

Xususan, statsionar tahlilda quyidagi tenglama yordamida iterativ hisoblash bajariladi.

\[ K_T^{(i)} \Delta T_{t=\infty}^{(i)} = F_{t=\infty} - K_T^{(i)} \Delta T_{t=\infty}^{(i)} \]
\[\begin{equation} T_{t=\infty}^{(i+1)} = T_{t=\infty}^{(i)} + \Delta{T}_{t=\infty}^{(i)} \label{eq:2.4.20} \end{equation}\]

Nostatsionar tahlilda vaqt inkrementi \(\Delta t\) ni tanlashda vaqt diskretlashtirishida noaniq sxema qo‘llangani sababli, odatda uning kattaligiga qat’iy barqarorlik cheklovi qo‘yilmaydi. Biroq vaqt inkrementi \(\Delta t\) juda katta bo‘lsa, iterativ hisoblashdagi yaqinlashish iteratsiyalari soni ortadi. Umuman, vaqt inkrementi \(\Delta t\) juda katta bo‘lsa, iteratsiyalar soni ko‘payadi. Amalda qoldiq vektorining kattaligi kuzatilib, yaqinlashish sekin bo‘lsa \(\Delta t\) kichraytiriladi, iteratsiyalar kam bo‘lsa \(\Delta t\) kattalashtiriladi; bunday avtomatik inkrement nazorati qo‘llanadi (→ tafsilotlar uchun Qadam nazorati ga qarang).

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