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Decarbonizing the built environment is crucial for achieving global sustainability goals, as buildings and infrastructure contribute significantly to carbon emissions. This study explores integrating direct air carbon capture, utilizing CaCO3-based technologies, into urban buildings through passive sustainable design. A computational framework was developed to optimize architectural design and enclosure geometry for enhanced passive airflow, using mass flow rate as a proxy for carbon absorption potential. Implemented within Rhino3D and Grasshopper using Ladybug and Eddy3D, the workflow integrates weather data and CFD simulation to compute segmented mass flow rates through stacked capture trays. The framework simplifies traditionally complex CFD processes by introducing a custom segmented mass-flow calculation approach that enables comparative performance assessment during early-stage design. Results confirm the validity of the proposed workflow, revealing that façade rotation can modify total mass flow by up to 96.5%, seasonal wind variability can cause airflow to range from approximately 8.5 kg/s in January to 169.5 kg/s in May in Seattle, and tower shadowing can reduce flow by up to 60.9%, demonstrating the strong influence of enclosure design and spatial configuration on passive carbon capture potential. This research establishes a performance-driven design framework that enables architectural geometry to actively enhance passive carbon capture integration, positioning building design as a measurable contributor to climate mitigation strategies.