In this paper, new constitutive equations for linear entangled polymer solutions and melts are derived from a recently proposed kinetic model (Fang et al. 2004) by using five closure approximations available in the literature. The simplest closure approximation considered is that due to Peterlin (1966). In this case, a mean-field-type Fokker-Planck equation underlying the evolution equation for an equilibrium averaged polymer segment orientation tensor is shown to be consistent with the fluctuation-dissipation theorem (Kubo et al. 1985). We compare the performance of the five new constitutive equations in their capacity to faithfully reproduce the predictions of the modified encapsulated FENE dumbbell model of Fang et al. (2004) for a number of shear and extensional flows. Comparisons are also made with the experimental data of Kahvand (1995) and Bhattacharjee et al. (2002, 2003). In the case of the Hinch-Leal and Bingham closures (Hinch and Leal 1976; Chaubal and Leal 1998) a combination with the quadratic closure of Doi (1981) is found to be necessary for stability in fast flows. The Hinch-Leal closure approximation, modified in this way, is found to outperform the other closures and its mathematical description is considerably simpler than that of the Bingham closure.