Solvents are key to many chemical and energy conversion processes. Solvents should be selected as part of process design, optimizing a process-level objective to account for the interactions between molecular properties and process performance. In this paper, we integrate the computer-aided molecular design of solvents with the design of heat-integrated processes for minimum utility demand. The process flowsheet is represented by thermodynamically accurate shortcut process models, encompassing the most common unit operations: extraction, distillation, absorption, and multiphase reaction. For each candidate solvent, we optimize the process considering heat integration and design solvents based on their process performance. All thermodynamic properties are predicted using quantum chemistry. The method is applied to two case studies: extraction-distillation and integrated carbon capture and utilization. In both studies, designed solvents improve process performance compared to literature benchmarks, where simpler heuristics lead to suboptimal choices. Thus, the results highlight the importance of integrating molecular and process design to achieve maximum process performance.