11 Boundary-Attached One-Dimensional Charring-Material Response

A recent CMPS extension couples the compressible gas-side wall solution to an independent one-dimensional material column attached to each active charring wall face. The CFD surface geometry remains fixed. The model predicts in-depth thermal response and virgin-to-char conversion; it does not, by itself, model oxidation, mechanical char removal, or geometric surface recession.

11.1 Material coordinate and conversion

Let \(x\in[0,L]\) measure distance into the material from the gas-facing surface. The conversion variable satisfies \[0\le\alpha(x,t)\le1,\] with \(\alpha=0\) virgin and \(\alpha=1\) fully converted char. A one-step Arrhenius reaction is \[\frac{\partial\alpha}{\partial t} =k_r(T)(1-\alpha)^n, \qquad k_r(T)=A\exp\left(-\frac{E_a}{RT}\right). \](11.1)

11.2 Conversion-dependent properties

The volumetric heat capacity and thermal conductivity are interpolated between virgin and char properties, \[\begin{aligned} C_V(\alpha) &=(1-\alpha)\rho_vc_{p,v}+\alpha\rho_cc_{p,c},\\ k_m(\alpha) &=(1-\alpha)k_v+\alpha k_c. \end{aligned}\](11.2) The in-depth energy equation is \[C_V(\alpha)\frac{\partial T}{\partial t} =\frac{\partial}{\partial x} \left(k_m(\alpha)\frac{\partial T}{\partial x}\right) -(\rho_v-\rho_c)\Delta h_p \frac{\partial\alpha}{\partial t}. \](11.3) The last term represents the modeled pyrolysis/decomposition enthalpy.

11.3 One-dimensional finite-volume discretization

Each wall face owns a material column discretized by one-dimensional finite volumes. The documented implementation uses boundary half cells and harmonic face conductivities. Conduction is treated implicitly. For a fixed temperature iterate, Eq. (11.1) is advanced analytically, and relaxed Picard iterations couple conversion and temperature. If a material substep fails, it is reduced and retried rather than accepting a non-finite or unbounded state.

The back face of the material column can be modeled as adiabatic, fixed temperature, or coupled to a solid-side condition according to the active boundary configuration.

11.4 Pyrolysis-gas production

The conversion increment produces a local pyrolysis-gas mass flux per unit wall area, \[\dot m''_p(t) =(\rho_v-\rho_c) \int_0^L\frac{\partial\alpha(x,t)}{\partial t}\,dx. \](11.4) The present model assumes that the generated gas escapes immediately from the material column; porous in-depth gas transport is not solved.

11.5 Film-theory blowing correction

The first-order influence of generated gas on gas-side convection is represented by a blowing correction. Using the unblown coefficient \(h_0\) from Eq. (10.7), define \[B_h=\frac{\dot m''_pc_{p,g}}{h_0}, \qquad \phi=2\lambda B_h,\] with \[\lambda= \begin{cases} 0.4, & \text{turbulent},\\ 0.5, & \text{laminar}. \end{cases}\] The film factor is \[F_B=\frac{\phi}{\exp(\phi)-1}, \qquad F_B\rightarrow1\quad\text{as}\quad\phi\rightarrow0. \](11.5) The corrected gas-side heat flux is \[q''_g =F_B\left[h_0(T_g-T_s)+q''_{\mathrm{visc}}\right], \qquad h_{\mathrm{eff}}=F_Bh_0. \](11.6) For positive blowing, the factor reduces the convective heat-transfer response.

11.6 Implicit CFD coupling

The wall/material evaluation is executed inside the implicit boundary assembly. AD therefore provides the local sensitivity of the accepted wall heat flux to the active CFD state. This is stronger than prescribing a fixed external heat-transfer coefficient because the gas-side boundary layer, wall temperature, material conversion, and blowing correction remain coupled through the current wall state.

For transient CFD, a material trial column starts from the previously accepted physical-time state and is committed only after the physical step is accepted. For steady CFD with a prescribed exposure time, the material state is reconstructed against that physical exposure rather than interpreting nonlinear CFD iterations as material time.

11.7 Engineering output measures

The model reports an end-time equivalent decomposition rate \[\dot s_{eq} =\frac{\dot m''_p}{\rho_v-\rho_c}\] and a cumulative equivalent decomposition depth \[s_{eq}(t)=\int_0^L\alpha(x,t)\,dx, \qquad m''_{p,\mathrm{cum}}=(\rho_v-\rho_c)s_{eq}(t). \](11.7) These are conversion measures. They must not be interpreted as the physical recession of the external wall while char is retained.

11.8 Current limitations

The current coupling applies the pyrolysis influence through the thermal blowing correction. Pyrolysis mass, momentum, energy, and species are not yet injected as conservative source/flux terms into the gas control volume. The model therefore does not represent the complete gas-side consequences of mass addition. Quantitative physical ablation also requires physics for oxidation, char removal, and geometric recession when those mechanisms are relevant.