Many industrial applications rely on real-time optimization to improve key performance indicators. In the case of unknown process characteristics, real-time optimization becomes challenging, particularly for the satisfaction of safety constraints. In this paper, we demonstrate the application of an adaptive and explorative real-time optimization framework to an industrial refrigeration process, where we learn the process characteristics through changes in process control targets and through exploration to satisfy safety constraints. We quantify the uncertainty in unknown compressor characteristics of the refrigeration plant by using Gaussian processes and incorporate this uncertainty into the objective function of the real-time optimization problem as a weighted cost term. We adaptively control the weight of this term to drive exploration. The results of our simulation experiments indicate the proposed approach can help to increase the energy efficiency of the considered refrigeration process, closely approximating the performance of a solution that has complete information about the compressor performance characteristics.