This paper focuses on multiscale dynamics occurring in steam supply systems. The dynamics of interest are originally described by a distributed-parameter model for fast steam flows over a pipe network coupled with a lumped-parameter model for slow internal dynamics of boilers. We derive a lumped-parameter model for the dynamics through physically-relevant approximations. The derived model is then analyzed theoretically and numerically in terms of existence of normally hyperbolic invariant manifold in the phase space of the model. The existence of the manifold is a dynamical evidence that the derived model preserves the slow-fast dynamics, and suggests a separation principle of short-term and long-term operations of steam supply systems, which is analogue to electric power systems. We also quantitatively verify the correctness of the derived model by comparison with brute-force simulation of the original model. * Y. Susuki is currently with and also with JST CREST, 4-1-8 Honcho, Kawaguchi, Saitama, 332-0012 Japan 1 In this paper, we newly present the unified description of multiscale dynamics of steam supply systems by proper scaling of governing equations in Sec. 3, derivation of inner-limit of the derived model in Sec. 4, and numerical simulations for phase-space analysis and for verification of the model in Sec. 5, all of which are not reported in [3,4].