In an early approach, we proposed a kinetic model with multiple translational temperature [K. Xu, H. Liu and J. Jiang, Phys. Fluids 19, 016101 (2007)]. Based on this model, the stress strain relationship in the Navier-Stokes (NS) equations is replaced by the translational temperature relaxation terms. The kinetic model has been successfully used in both continuum and near continuum flow computations. In this paper, we will further validate the multiple translational temperature kinetic model to flow problems in multiple dimensions. First, a generalized boundary condition incorporating the physics of Knudsen layer will be introduced into the model. Second, the direct particle collision with the wall will be considered as well for the further modification of particle collision time, subsequently a new effective viscosity coefficient will be defined. In order to apply the kinetic model to near continuum flow computations, the gas-kinetic scheme will be constructed. The first example is the pressure-driven Poiseuille flow at Knudsen number 0.1, where the anomalous phenomena between the results of the NS equations and the Direct Simulation Monte Carlo (DSMC) method will be resolved through the multiple temperature model. The so-called Burnett-order effects can be captured as well by algebraic temperature relaxation terms. Another test case is the force-driven Poiseuille flow at various Knudsen numbers. With the effective viscosity approach and the generalized second-order slip boundary condition, the Knudsen minimum can be accurately obtained. The current study indicates that it is useful to use multiple temperature concept to model the non-equilibrium state in near continuum flow limit. In the continuum flow regime, the multiple temperature model will automatically recover the single temperature NS equations due to the efficient energy exchange in different directions.