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This work explores the model predictive controller design of the continuous pulp digester process consisting of the co-current zone and counter-current zone modelled by a set of nonlinear coupled hyperbolic partial differential equations (PDE). The distributed parameter system of interest is not spectral and slow-fast dynamic separation does not hold. To address this challenge, the nonlinear continuous-time model is linearized and discretized in time utilizing the Cayley-Tustin discretization framework, which ensures system theoretic properties and structure preservation without spatial discretization or model reduction. The discrete model is used in the full state model predictive controller design, which is augmented by the Luenberger observer design to achieve the output constrained regulation. Finally, a numerical example is provided to demonstrate the feasibility and applicability of the proposed controller designs.