The paper is intented to show a new, state space, discrete, non integer order model of a one-dimensional heat transfer process. The proposed model derives directly from time continuous, state space model and it uses the discrete Grünwald-Letnikov operator to express the fractional order difference with respect to time. Stability and spectrum decomposition for the proposed model are recalled, the accuracy and convergence are analyzed too. The convergence of the proposed model does not depend on parameters of heater and measuring sensors. The dimension of the model assuring stability and predefined rate of convergence and stability is estimated. Analytical results are confirmed by experiments.