Abstract The spread of fire across the ceiling of a large room (or long corridor) is modeled as wind-aided flame spread along a horizontal char-forming thick slab, in the presence of significant convective, diffusive, and radiative transport. The goal is to predict the rate of streamwise advance of the site on the solid-gas interface at which the pristine solid undergoes endothermic degradation to a combination of (I) a porous carbonaceous heat-retaining matrix, and (2) a mixture of (partially combustible) vapors that move through the matrix to the outer gas. This rate of advance of the thermal-degradation site is sought as a function of normally available data concerning the thermodynamic and physical properties of the solid; the thermodynamic and dynamic state of the hot vitiated bulk gas that abruptly starts, and then continues, to flow over the slab; and the initial thermodynamic state of the slab. Downwind of the interfacial degradation site, hot product gases flowing over the slab preheat it from its ambient to its outgassing state; upwind of the degradation site, formation of a thickening char layer between the deep pristine solid and the bulk gas occurs, though, far enough upstream, the char layer thins because the carbonaceous matrix is eroded under surface attack. The degradation site is thus a “separatrix” whose time-varying position, to be found in the course of solution, conveniently characterizes the extent of slab “involvement” this moving separatrix implies a boundary/initial-value problem of unconventional, but tractable, Stefan type. A nonlinear, unsteady, two-spatial-dimension treatment in the Shvab-Zeldovich approximation entails boundary-layer simplification in the manner of Prandtl, convective-transport simplification in the manner of Oseen, and thin-flame simplification in the manner of Burke and Schumann. The char layer plays an important role because of its thermal resistance and its heat retention; and radiative transfer, is also important, via bulk-gas cooling, solid-surface emission, and near-flame radiation. It is not surprising that an analysis of the time-independent counterpart of this phenomenon in the presence of this plethora of mechanisms† has not yet been carried out, and, accordingly, here in Part I, we formulate the time-dependent problem in anticipation of its treatment in Part II and we solve the steady-state problem to get the insight it provides, to establish the methodology, and to obtain information which is needed in the postulation of the boundary conditions of the time-dependent phenomenon.