Mlinganyo wa upitishaji joto¶

Sura hii inapanga governing equation ya upitishaji joto katika kiwango cha continuum na masharti ya mpaka yanayoshughulikiwa na FrontISTR. Udiscretishaji wa muda na FEM unaelezwa katika sehemu ya discretization (L3d) na algoriti ya utatuzi (L3e: solution_methods/05_transient_heat).

Governing equation¶

Mlinganyo wa upitishaji joto katika continuum ni kama ifuatavyo.

\[\begin{equation} \rho c \frac{\partial T}{\partial t} = \frac{\partial}{\partial x}\left(k_x \frac{\partial T}{\partial x}\right) + \frac{\partial}{\partial y}\left(k_y \frac{\partial T}{\partial y}\right) + \frac{\partial}{\partial z}\left(k_z \frac{\partial T}{\partial z}\right) + Q \label{eq:gov_he_main} \end{equation}\]

Hapa, \(\rho=\rho(x)\) ni mass density, \(c=c(x,T)\) ni specific heat, \(T=T(x,t)\) ni temperature, \(k=k(x,T)\) ni thermal conductivity, \(Q=Q(x,T,t)\) ni heat generation, \(x\) ni nafasi, na \(t\) ni muda. \(k_x, k_y, k_z\) ni thermal conductivity katika kila mwelekeo kwa upitishaji joto anisotropic.

Kwa sifa maalum za joto (utegemezi wa temperature na anisotropy), rejelea Sifa za joto.

Masharti ya mpaka¶

Tuchukulie domain inayozingatiwa kuwa \(S\) na mpaka wake kuwa \(\Gamma\). Tukidhani kwamba katika kila sehemu ya \(\Gamma\) limepewa sharti la mpaka la aina ya Dirichlet au Neumann, masharti ya mpaka huwa kama ifuatavyo.

\[\begin{equation} T = T_1(x,t), \; x \in \Gamma_1 \label{eq:gov_he_dirichlet} \end{equation}\]

\[\begin{equation} k \frac{\partial T}{\partial n} = q(x,T,t), \; x \in \Gamma_2 \label{eq:gov_he_neumann} \end{equation}\]

Hapa, maumbo ya kazi ya \(T_1\) na \(q\) yanachukuliwa kuwa yanajulikana. \(q\) ni heat flux inayotoka kwenye mpaka. Heat flux \(q\) kutoka kwenye mpaka inajumuisha sehemu tatu zifuatazo.

\[\begin{equation} q=-q_s+q_c+q_r \label{eq:gov_he_flux} \end{equation}\]

\[\begin{equation} q_s=q_s(x,t) \label{eq:gov_he_qs} \end{equation}\]

\[\begin{equation} q_c=hc(T-Tc) \label{eq:gov_he_qc} \end{equation}\]

\[\begin{equation} q_r=hr(T^4-Tr^4) \label{eq:gov_he_qr} \end{equation}\]

Hapa \(q_s\) ni distributed heat flux, \(q_c\) ni heat flux kutokana na convection, na \(q_r\) ni heat flux kutokana na radiation.

Ambapo \(Tc=Tc(x,t)\) ni ambient temperature ya convective heat transfer, \(hc=hc(x,t)\) ni convective heat-transfer coefficient, \(Tr=Tr(x,t)\) ni ambient temperature ya radiative heat transfer, \(hr=\varepsilon \sigma F = hr(x,t)\) ni radiative heat-transfer coefficient, \(\varepsilon\) ni emissivity, \(\sigma\) ni Stefan-Boltzmann constant, na \(F\) ni view factor.

Kwa jinsi ya kubainisha haya katika faili ya ingizo, rejelea Masharti ya mpaka na mizigo katika sehemu ya vipengele.

Weak form¶

Mpaka wa coupling na uchanganuzi wa miundo¶

Vipengee vinavyohusiana¶

Upitishaji joto usio steady — Udiscretishaji wa muda na nonlinear iteration
Sifa za joto — Uundaji wa data ya sifa na utegemezi wa temperature
Shape functions na makadirio ya vipengele vya mwisho — Mfumo wa FEM discretization
Rejea ya vipengele Masharti ya mpaka na mizigo — Ingizo la masharti ya mpaka kwa uchanganuzi wa upitishaji joto