We　consider　non-isentropic　Euler-Maxwell　equations　with　relaxation　times　(small　physical　parameters)　arising　in　the　models　of　magnetized　plasma　and　semiconductors.　For　smooth　periodic　initial　data　sufficiently　close　to　constant　steady-states,　we　prove　the　uniformly　global　existence　of　smooth　solutions　with　respect　to　the　parameter,　and　the　solutions　converge　global-in-time　to　the　solutions　of　the　energy-transport　equations　in　a　slow　time　scaling　as　the　relaxation　time　goes　to　zero.　We　also　establish　error　estimates　between　the　smooth　periodic　solutions　of　the　non-isentropic　Euler-Maxwell　equations　and　those　of　energy-transport　equations.　The　proof　is　based　on　stream　function　techniques　and　the　classical　energy　method　but　with　some　new　developments.　(c)　2024　Elsevier　Inc.　All　rights　are　reserved,　including　those　for　text　and　data　mining,　AI　training,　and　similar　technologies.