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.　©　2024　Elsevier　Inc.