In a hot strip mill, the strip is cooled by spraying water from top and bottom on the runout table (ROT), before the strip is coiled. The desired mechanical properties and metallurgical structure of the strip are achieved by controlling the cooling rate and temperature of the strip on the ROT. In this paper, we propose a design method of model predictive control for the temperature of the strip whose mathematical model is described by a nonlinear partial differential equation (PDE). We provide a novel framework in which a fast numerical algorithm for solving nonlinear optimization problem is applicable to the model predictive control for a class of nonlinear PDEs. The effectiveness of the proposed controller and estimator is verified by numerical simulations.