Bioreactors are essential technologies for the bioeconomy, as they enable the main unit operations of bioenergy and circularity bioprocesses. These include the production of biofuels, biobased materials and basic chemicals from bioresources. The design of industrial-scale bioreactors is a challenging problem due to the multiscale and multiphysics nature of the system. However, it is crucial to have reliable methods to scale up bioprocesses from lab to industry to accelerate the deployment of bioeconomy as a solution for climate change and other environmental issues. The scale-up of bioreactors is difficult for several reasons: The uniqueness of each reactor, which depends on its geographical location (e.g., weather and bioresources); the complex operation due to the heterogeneity of the feed, the microbial sensitivity, and the non-steady behavior; the large volume production requirements; and the urgency in the implementation. We propose a scale-up methodology based on a Representative Elementary Volume (REV) of the bioreactor and show how this can facilitate the design and scale-up of novel bioreactors relevant for the bioeconomy. The methodology is designed to: Combine experimental and numerical approaches; allow the study of dynamic behavior; be highly computationally efficient; and consider the multiscale and multiphysics features of the process. We use a reactive extrusion reactor as an example to illustrate the methodology, which consists of: selecting an REV, building a Reduced Order Model (ROM) using explainable Machine Learning methods, defining similarity criteria based on Probability Density Functions (PDFs) of dimensionless numbers, and modeling the transient performance of the bioreactor based on process system modeling using REV ROMs.