The present work aims at applying the ideas on the analysis of improved lumped-parameter model for transient heat conduction in a slab with temperature-dependent thermal conductivity. The transient temperature is found to depend on various model parameters, namely, Biot number, heat source parameter and time. Polynomial Approximation Method (PAM) has been possible to derive a unified relation for the transient thermal behaviour of solid (slab and tube) with both internal generation and boundary heat flux. In all the cases, a closed form solution is obtained between temperature, Biot number, heat source parameter and time. An improved lumped parameter model has been adopted through two point Hermite approximations for integrals. For linearly temperature-dependent thermal conductivity, it is shown by comparison with numerical solution of the original distributed parameter model that the higher order lumped model (H1,1/H0,0 Approximations) yields significant improvement of average temperature prediction over the classical lumped model. A unified Biot number limit depending on a single dimensionless parameter is given both for cooling and heating processes. The result of the present analysis has been compared with earlier numerical and analytical results. A good agreement has been obtained between the present prediction and the available results.