In the case of conjugate heat transfer, the dissipation rate associated with the temperature variance is discontinuous at the fluid-solid interface. The discontinuity satisfies a compatibility condition involving the fluid-solid thermal diffusivity and conductivity ratios and the relative contribution to the dissipation rate of its wall-normal part. The present analysis is supported by the Direct Numerical Simulations of an incompressible channel flow at a Reynolds number, based on the friction velocity, of 150, a Prandtl number of 0.71 and several values of fluid-solid thermal diffusivity and conductivity ratios.