Comparison theorems for exothermally reacting solids in conjugate solid-fluid systems are given. The solids are assumed to be of arbitrary shape and the fluid assumed to be well mixed and in continuous flow. In general the concentration and thermal profiles in the solid are intimately connected with those variables in the fluid leading to integral boundary conditions.The comparison theorems for a zeroth-order reaction show the existence and dependence of basic solutions on system parameters and geometry or domain size. Similar results are also obtained for the ignition point during critical conditions. We have not performed any physical or numerical experiments; only analytical perturbation methods, monotone iteration and the Reynolds transport theorem are used. Comparison with numerical calculations of earlier workers is given for some of our results. Our theorems have been made possible because an unusual inner product has rendered the eigenvalue problem of ignition (nonlinear in the eigenvalue parameter) self-adjoint. We should like to point out that the domain-dependence proof will allow us to consider simpler one-dimensional problems instead of complex three-dimensional problems. © 1988 Oxford University Press.