Direct air capture (DAC) of carbon dioxide is a promising technology to enable climate change mitigation. The liquid solvent DAC (LSDAC) process is one of the leading technologies being piloted. However, LSDAC uses a high-temperature regeneration process which requires a lot of thermal energy. Although current LSDAC designs incorporate pre-heat cyclones and a heat recovery steam generator to enable heat recovery, these do not maximize the use of the heat in the products of calcination. In this paper, a linear optimization model is developed to minimize energy cost in a LSDAC that is powered by renewable energy and natural gas. First, the material flow network is modified to include a heat exchanger (HX) and water supply to a proton exchange membrane (PEM) electrolyser. Mass and energy balance constraints are then developed to include the water flow as well as the energy balance at the PEM and the HX. Results show that about 911 tonnes of hydrogen could be produced over 336 hours of operation using a 136MW PEM. Further analysis reveals that hydrogen production is only prioritized if the value is higher than the cost of natural gas.