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Environmental concerns regarding global warming and ozone depletion urge towards sustainable solutions for satisfying the increasing cooling demand. Adsorption cooling technology could form part of the solution since it can be driven by solar energy and industrial or vehicular waste heat, as well as it employs non ozone-destructive refrigerants. However, its low performance hinders its extensive development and commercialization. The design of the adsorption reactor is crucial for its performance improvement, since its inherent cyclic operation imposes a compromise between the Specific Cooling Power and the Coefficient of Performance. A generalized three-dimensional computational model based on unstructured meshes is presented, capable to simulate all potential geometries. Dynamic conjugate simulations of the packed bed and the heat exchanger allow to study the latter's influence on the reactor performance. A parametric study of five reactor geometries was conducted, demonstrating quantitatively the strong impact of the solid volume fraction, fin length and fin thickness on the performance. Within the studied range, the Specific Cooling Power is maximized for the highest solid volume fraction and for the lowest fin thickness and fin length. The effect of the adsorbed mass spatial distribution on the desorption phase is discussed. A sensitivity analysis exhibits the importance of the heat transfer coefficient between the two domains. Copper and aluminium are compared as heat exchanger materials, revealing that the former performs more effectively, although the difference is appreciable only for longer fin lengths. The presented numerical model can be employed for improving the design of adsorption packed bed reactors.