Laminar fluid flow and heat transfer phenomena in straight ducts with injection of the same fluid through one or more side walls are often encountered in vapor passages of the evaporator sections of loop heat pipes, vapor-chamber heat spreaders, and two-phase thermosyphons, and also in fluid-flow passages of fuel cells. The design and optimization of these devices are facilitated by computer simulations based on cost-effective hybrid numerical models of the thermofluid phenomena, involving coupled numerical solutions of multidimensional and quasi-one-dimensional mathematical models, with inputs in the latter of correlations for pressure drops and heat transfer, based on either empirical or numerically generated data. The overall goal in this paper is to propose an efficient numerical model for generating such data for straight vapor ducts of the evaporator section, when the prevailing conditions lead to effectively fully developed fluid flow and heat transfer. In this fully developed region, suitably nondimensionalized velocity and temperature fields become invariant in the axial (main flow) direction; and the product of friction factor (Darcy or Fanning) and axial-flow Reynolds number, and a dimensionless bulk temperature, attain constant values for each set of dimensionless parameters. Several works in the published literature have demonstrated the existence of such fully developed fluid flow and heat transfer phenomena using solutions of mathematical models that include the developing region. In this work, a novel dimensionless mathematical model that allows predictions of these phenomena directly in the fully developed region is proposed and solved using a finite volume method. A pattern-preserving grid-refinement scheme and a generalized Richardson extrapolation procedure are used to estimate grid-independent solutions. Verification is done by comparing the results with previously published results yielded by numerical solutions of mathematical models of developing fluid flow and heat transfer phenomena from the inlet plane to the fully developed region.